Question 1

If $2^x = \sqrt[3]{32}$, then $x$ is equal to

Accepted Answer

Concept & Formula: This problem is a fundamental test of translating radical notation (roots) into fractional exponents. Once the root is a fraction, you simply express the internal number as a prime base to equate both sides. The core conversion rule is: \sqrt[n]{x^m} = x^{(m/n)} Step-by-Step Solution: • Given: 2^x = \sqrt[3]{32} • Calculation: First, evaluate the number inside the cube root. We need to express 32 as a power of the base 2. We know that 2 × 2 × 2 × 2 × 2 = 32. Therefore, 32 = 2^5. Substitute this back into the radical expression: 2^x = \sqrt[3]{2^5} Convert the cube root into a fractional exponent. A cube root corresponds to the denominator 3 in a fractional power: 2^x = (2^5)^{(1/3)} Multiply the exponents using the power of a power rule: 2^x = 2^{(5/3)} Since the bases on both sides of the equation are now exactly 2, we can drop the bases and directly equate the exponents: x = (5/3) Exam Strategy & Shortcut: Direct Translation: Memorize the prime powers up to 2^6 and 3^5. As soon as you read \sqrt[3]{32}, you should mentally translate 32 to 2^5, and the cube root to a division by 3 on the exponent, immediately yielding (5/3). This is a 5-second mental math problem. Common Pitfall: A frequent trap is confusing the numerator and denominator when converting roots. Students sometimes mistakenly write (2^5)^{(1/3)} as 2^{(3/5)}, leading them to confidently pick the incorrect option (c). Remember: Power is on top, Root is on the bottom. Final Answer: Therefore, the correct answer is 5/3.

Question 2

If $2^x 	imes 8^{\frac{1}{5}} = 2^{\frac{1}{5}}$, then $x$ is equal to

Accepted Answer

Concept & Formula: The core concept is to convert all composite bases into prime bases so that the laws of indices can be applied uniformly. When dealing with bases like 2 and 8, recognize that 8 = 2^3. The necessary laws of indices are: (x^m)^n = x^{mn} x^m × x^n = x^{m+n} Step-by-Step Solution: • Given: 2^x × 8^{(1/5)} = 2^{(1/5)} • Calculation: Convert the base 8 into a power of 2. We know that 8 = 2^3. Substitute 2^3 into the equation: 2^x × (2^3)^{(1/5)} = 2^{(1/5)} Apply the power of a power rule by multiplying the exponents: 2^x × 2^{(3/5)} = 2^{(1/5)} Apply the product rule by adding the exponents on the left-hand side: 2^{x + (3/5)} = 2^{(1/5)} Since the bases on both sides are identical, equate the exponents: x + (3/5) = (1/5) Isolate x by subtracting (3/5) from both sides: x = (1/5) - (3/5) x = -(2/5) Exam Strategy & Shortcut: Exponent-Only Equation: Mentally translate 8 to 2^3 and skip writing the bases entirely. Immediately write down the linear equation for the exponents: x + (3/5) = (1/5). Solving x = (1/5) - (3/5) = -(2/5) takes less than 10 seconds and prevents transcription errors. Common Pitfall: A frequent error is misapplying the product rule and multiplying the exponents instead of adding them, which leads to x \cdot (3/5) = (1/5), incorrectly yielding x = (1/3). Always remember that multiplying identical bases means adding their exponents. Final Answer: Therefore, the correct answer is -2/5.

Question 3

$\frac{2^{n+4} - 2(2^n)}{2(2^{n+3})}$ when simplified is

Accepted Answer

Concept & Formula: When simplifying algebraic fractions with variable exponents, the goal is to factor out the smallest common power of the base from both the numerator and the denominator so they can be canceled out. The key laws of indices required are: x^m × x^n = x^{m+n} x^{m+n} = x^m × x^n Step-by-Step Solution: • Given: \frac{2^{n+4} - 2(2^n)}{2(2^{n+3})} • Calculation: First, break down the exponents in the numerator using the product rule in reverse (x^{m+n} = x^m \cdot x^n): Numerator = 2^n \cdot 2^4 - 2^1 \cdot 2^n Factor out the common term, 2^n, from the numerator: 2^n(2^4 - 2^1) = 2^n(16 - 2) = 2^n \cdot 14 Now, simplify the denominator. Remember that 2 is 2^1: Denominator = 2^1 \cdot 2^{n+3} = 2^{1 + n + 3} = 2^{n+4} Break this down to isolate 2^n: 2^{n+4} = 2^n \cdot 2^4 = 2^n \cdot 16 Substitute the simplified numerator and denominator back into the fraction: (2^n \cdot 14/2^n \cdot 16) Cancel out the common 2^n term from the top and bottom: (14/16) Reduce the fraction to its lowest terms by dividing by 2: (7/8) Exam Strategy & Shortcut: Value Substitution (Plug-in Method): Since the options primarily consist of constants (and one of them is the correct answer), this means the expression must hold true for any arbitrary value of n. Assume n = 0 for the easiest calculation: Numerator: 2^{(0+4)} - 2(2^0) = 2^4 - 2(1) = 16 - 2 = 14 Denominator: 2(2^{(0+3)}) = 2(2^3) = 2(8) = 16 Result: (14/16) = (7/8). This perfectly matches option (d) in just a few seconds of basic arithmetic. Common Pitfall: A standard error occurs when students try to subtract exponents across subtraction signs in the numerator, incorrectly attempting to apply division rules to individual terms before factoring. Always factor completely before attempting to cancel terms across a division line. Final Answer: Therefore, the correct answer is (7/8).

Question 4

$(0.04)^{-1.5} = x$

Accepted Answer

Concept & Formula: This problem tests two key concepts: handling negative exponents and converting decimals to fractions. A negative exponent indicates the reciprocal of the base. a^{-m} = (1/a^m) = ((1/a))^m Step-by-Step Solution: • Given the expression: (0.04)^{-1.5} • First, convert the decimal base into a fraction to make it easier to manipulate. • 0.04 = (4/100) = (1/25) • Substitute this back into the expression: ((1/25))^{-1.5} • Apply the negative exponent rule to flip the fraction and make the exponent positive: (25)^{1.5} • Convert the decimal exponent to a fraction: 1.5 = (3/2). • The expression is now: (25)^{(3/2)} • This means taking the square root of 25 and then cubing the result. Since 25 = 5^2: • (5^2)^{(3/2)} = 5^{2 × (3/2)} = 5^3 • Calculate 5^3 = 5 × 5 × 5 = 125. Exam Strategy & Shortcut: Read 0.04 as (1/25). A negative power flips it to 25. A power of 1.5 means "the number times its square root". So, 25 × \sqrt{25} = 25 × 5 = 125. This avoids calculating 5^3 and is extremely fast to execute mentally. Common Pitfall: A common mistake is mishandling the negative sign in the exponent. Some might mistakenly multiply the base by a negative value or fail to invert the fraction properly, resulting in a tiny decimal answer rather than the correct whole number. Final Answer: Therefore, the correct answer is 125.

Question 5

$2^{3.6} 	imes 4^{3.6} 	imes 4^{3.6} 	imes (32)^{2.3} = (32)^{x}$

Accepted Answer

Concept & Formula: When multiple distinct bases share the exact same exponent, apply the power of a product rule in reverse to combine them into a single base. a^n × b^n × c^n = (a × b × c)^n Step-by-Step Solution: • Given: 2^{3.6} × 4^{3.6} × 4^{3.6} × (32)^{2.3} = (32)^x • Observe the first three terms. They all share the identical exponent of 3.6. • Apply the rule to combine the bases: • (2 × 4 × 4)^{3.6} × (32)^{2.3} = (32)^x • Calculate the new combined base: 2 × 4 = 8, and 8 × 4 = 32. • Substitute this back: • (32)^{3.6} × (32)^{2.3} = (32)^x • Now, the equation features identical bases. Add the exponents on the left side: • (32)^{3.6 + 2.3} = (32)^x • (32)^{5.9} = (32)^x • Equate the exponents: • x = 5.9 Exam Strategy & Shortcut: Before immediately converting everything to prime base 2, look for structural patterns. Spotting that 2 × 4 × 4 = 32 and that they all share the 3.6 exponent allows you to instantly collapse three terms into (32)^{3.6}. The problem then becomes a trivial mental addition: 3.6 + 2.3 = 5.9. Common Pitfall: The most common inefficient approach is converting all bases to 2 (e.g., making the left side 2^{3.6} × 2^{7.2} × 2^{7.2}). While mathematically sound, it involves larger decimals and takes significantly more time, increasing the risk of calculation errors. Final Answer: Therefore, the correct answer is 5.9.

Question 6

$3^{3.5} 	imes 21^2 	imes 42^{2.5} \div 2^{2.5} 	imes 7^{3.5} = 21^x$

Accepted Answer

Concept & Strategy: When faced with multiple different bases, look for relationships that allow you to combine them into the target base (in this case, 21). Use the power of a quotient and power of a product rules. (a^n/b^n) = ((a/b))^n a^n × b^n = (a × b)^n Step-by-Step Solution: • The target base on the right-hand side is 21. Our goal is to manipulate the left side to exclusively feature base 21. • First, address the division part of the expression: 42^{2.5} \div 2^{2.5}. • Since the exponents are the same, divide the bases: ((42/2))^{2.5} = 21^{2.5}. • Next, look at the terms 3^{3.5} and 7^{3.5}. They also share the same exponent. • Combine them by multiplying the bases: (3 × 7)^{3.5} = 21^{3.5}. • Now, substitute these simplified terms back into the original equation. • 21^{3.5} × 21^2 × 21^{2.5} = 21^x • Since all bases are now identical (21) and are being multiplied, simply add their exponents. • 3.5 + 2 + 2.5 = 8 • Therefore, 21^8 = 21^x, which means x = 8. Exam Strategy & Shortcut: Scan the equation and immediately identify the target base: 21. Look for pairs that form 21. Mentally pair 42 \div 2 to get 21^{2.5}, and 3 × 7 to get 21^{3.5}. You are left with exponents 3.5, 2, and 2.5. Add them up in your head: 3.5 + 2.5 = 6, plus 2 = 8. Done in 10 seconds without writing a single step. Common Pitfall: A major trap is blindly following BODMAS from left to right without looking for grouping opportunities, or incorrectly trying to apply laws of indices to terms with different bases and different exponents. Always unify the bases first. Final Answer: Therefore, the correct answer is 8.

Question 7

$8^7 	imes 2^6 \div 8^{2.4} = 8^x$

Accepted Answer

Concept & Formula: This problem tests your ability to manipulate indices using both multiplication (addition of exponents) and division (subtraction of exponents) rules. a^m × a^n \div a^p = a^{m+n-p} Step-by-Step Solution: • Observe the target base on the right side of the equation, which is 8. • Most terms are already in base 8. We only need to convert 2^6 into base 8 to make calculation seamless. • Recall that 8 = 2^3. Therefore, 2^6 = (2^3)^2 = 8^2. • Substitute this back into the original equation: • 8^7 × 8^2 \div 8^{2.4} = 8^x • Apply the laws of indices. Add the exponents for multiplication and subtract the exponent for division: • 8^{7 + 2 - 2.4} = 8^x • Perform the arithmetic on the exponents: • 7 + 2 = 9 • 9 - 2.4 = 6.6 • So, 8^{6.6} = 8^x. Equating the exponents gives x = 6.6. Exam Strategy & Shortcut: Keep your focus strictly on the target base (8). Do not waste time converting 8^7 and 8^{2.4} into base 2, as that creates unnecessarily large numbers and decimals. Just swap 2^6 for 8^2. The math then becomes a trivial mental calculation: 7 + 2 - 2.4 = 6.6. Common Pitfall: Converting all terms to base 2 is a common trap. While mathematically valid (yielding 2^{21} × 2^6 \div 2^{7.2} = 2^{3x}), it takes significantly longer and involves solving 27 - 7.2 = 3x, which introduces more opportunities for arithmetic errors under time pressure. Final Answer: Therefore, the correct answer is 6.6.

Question 8

$25^{2.7} 	imes 5^{4.2} \div 5^{5.4} = 25^{x}$

Accepted Answer

Concept & Formula: The problem tests the fundamental Laws of Indices, specifically converting different bases to a common prime base to equate exponents. The key formulas used are: (a^m)^n = a^{m × n} a^m × a^n = a^{m+n} a^m \div a^n = a^{m-n} Step-by-Step Solution: • Step 1: Convert all bases to the common prime base of 5. 25^{2.7} = (5^2)^{2.7} = 5^{5.4} The Right Hand Side (RHS) is 25^x = (5^2)^x = 5^{2x} • Step 2: Rewrite the full equation using the common base. 5^{5.4} × 5^{4.2} \div 5^{5.4} = 5^{2x} • Step 3: Apply the laws of multiplication and division of indices. 5^{5.4 + 4.2 - 5.4} = 5^{2x} 5^{4.2} = 5^{2x} • Step 4: Equate the exponents since the bases are equal. 2x = 4.2 x = 2.1 Exam Strategy & Shortcut: Notice that 25^{2.7} is exactly 5^{5.4}. When you see 5^{5.4} × \dots \div 5^{5.4}, the first and last terms on the Left Hand Side (LHS) immediately cancel each other out (i.e., a × b \div a = b). This instantly simplifies the LHS to 5^{4.2}. Equating this to 25^x (which is 5^{2x}) directly gives 2x = 4.2, or x = 2.1. This mental shortcut saves crucial seconds. Common Pitfall: A frequent error is forgetting to convert the target base (25) to the common base (5). Students might calculate the LHS exponent as 4.2 and incorrectly assume x = 4.2, failing to account for the 25^x turning into 5^{2x}. Final Answer: Therefore, the correct answer is None of these.

Question 9

$8^{2.4} 	imes 2^{3.7} \div (16)^{1.3} = 2^{x}$

Accepted Answer

Concept & Formula: This problem requires applying the Laws of Indices to simplify an expression by finding a common prime base. The essential exponent rules are: (a^m)^n = a^{m × n} a^m × a^n \div a^p = a^{m+n-p} Step-by-Step Solution: • Step 1: Identify the lowest common base, which is 2. Convert all other bases (8 and 16) to powers of 2. 8^{2.4} = (2^3)^{2.4} = 2^{7.2} (16)^{1.3} = (2^4)^{1.3} = 2^{5.2} • Step 2: Substitute these converted values back into the original equation. 2^{7.2} × 2^{3.7} \div 2^{5.2} = 2^x • Step 3: Apply the multiplicative and divisional laws of indices to combine the exponents on the Left Hand Side (LHS). 2^{7.2 + 3.7 - 5.2} = 2^x 2^{10.9 - 5.2} = 2^x 2^{5.7} = 2^x • Step 4: Equate the powers since the bases on both sides are identical. x = 5.7 Exam Strategy & Shortcut: To solve this rapidly, skip writing down the intermediate base steps. Mentally multiply the given powers by their base's relation to 2 (e.g., multiply 8's power by 3, and 16's power by 4). Just write down the final exponent string: 3(2.4) + 3.7 - 4(1.3) = 7.2 + 3.7 - 5.2. This avoids rewriting the base 2 repeatedly and directly yields 5.7. Common Pitfall: Students often mistake the division sign for a subtraction sign or misapply the division rule by dividing the exponents instead of subtracting them. Always remember that a^m \div a^n means subtracting the exponent n from m. Final Answer: Therefore, the correct answer is 5.7.

Question 10

$(0.04)^{2} \div (0.008) 	imes (0.2)^{6} = (0.2)^{x}$

Accepted Answer

Concept & Formula: The problem tests your ability to recognize decimal powers and apply the Laws of Indices. It requires converting varying decimal bases into a uniform decimal base. The governing rules are: (a^m)^n = a^{m × n} a^m \div a^n × a^p = a^{m-n+p} Step-by-Step Solution: • Step 1: Observe that the target base on the Right Hand Side is 0.2. Express the other decimals as powers of 0.2. 0.04 = (0.2)^2 0.008 = (0.2)^3 • Step 2: Substitute these into the original terms. (0.04)^2 = ((0.2)^2)^2 = (0.2)^4 The equation becomes: (0.2)^4 \div (0.2)^3 × (0.2)^6 = (0.2)^x • Step 3: Apply the laws of indices from left to right. (0.2)^{4 - 3 + 6} = (0.2)^x (0.2)^{1 + 6} = (0.2)^x (0.2)^7 = (0.2)^x • Step 4: Equate the exponents. x = 7 Exam Strategy & Shortcut: Instead of working with decimals which can be prone to error, you can focus strictly on the significant digit (2) and its place value. Recognize 0.04 as 2^2 shifted two decimal places, which aligns with (0.2)^2. Immediately write the exponent arithmetic: 2 × 2 - 3 + 6 = 7. By skipping the formal notation and mapping decimals to powers of 0.2, you arrive at x = 7 in just seconds. Common Pitfall: A major pitfall is applying the multiplication before the division, effectively calculating (0.2)^4 \div ((0.2)^3 × (0.2)^6) = (0.2)^{4 - 9} = (0.2)^{-5}. Always adhere strictly to left-to-right processing for operators of the same precedence (Division and Multiplication). Final Answer: Therefore, the correct answer is None of these.

Question 11

$(25)^{7.5} 	imes (5)^{2.5} \div (125)^{1.5} = 5^{x}$

Accepted Answer

Concept & Formula: This problem requires converting various composite bases into a common prime base to simplify the expression using the Laws of Indices. The key formulas are: (a^m)^n = a^{m × n} a^m × a^n = a^{m+n} a^m \div a^n = a^{m-n} Step-by-Step Solution: • Step 1: Identify the target base from the Right Hand Side, which is 5. Convert 25 and 125 to powers of 5. 25^{7.5} = (5^2)^{7.5} = 5^{15} 125^{1.5} = (5^3)^{1.5} = 5^{4.5} • Step 2: Substitute these simplified terms back into the original equation. 5^{15} × 5^{2.5} \div 5^{4.5} = 5^x • Step 3: Apply the laws of indices for multiplication (addition of exponents) and division (subtraction of exponents). 5^{15 + 2.5 - 4.5} = 5^x • Step 4: Simplify the exponent. 15 + 2.5 - 4.5 = 13 5^{13} = 5^x • Step 5: Equate the exponents. x = 13 Exam Strategy & Shortcut: Skip writing out the base conversions. Map each given base directly to its power of 5 (25 is 5^2, 125 is 5^3). Multiply these powers by the given exponents mentally: 2 × 7.5 = 15 and 3 × 1.5 = 4.5. Then, just write the final exponent arithmetic: 15 + 2.5 - 4.5 = 13. Common Pitfall: Students sometimes add all exponents blindly or misplace the decimal point during multiplication (e.g., 3 × 1.5 = 45 instead of 4.5), completely throwing off the final result. Final Answer: Therefore, the correct answer is 13.

Question 12

$(64x^3 \div 27a^{-3})^{-\frac{2}{3}}$

Accepted Answer

Concept & Formula: This problem tests your mastery of the Laws of Indices, specifically dealing with negative exponents and fractional powers. The key exponent rules required are: a^{-m} = (1/a^m) ((a/b))^{-m} = ((b/a))^m (a^m)^n = a^{m × n} Step-by-Step Solution: • Step 1: Simplify the expression inside the parentheses by removing the negative exponent on the variable a. 64x^3 \div 27a^{-3} = (64x^3/27a^{-3)} Since a^{-3} is in the denominator, it moves to the numerator as a^3: (64x^3 a^3/27) • Step 2: Apply the negative fractional exponent -(2/3) to the entire fraction. A negative exponent flips the fraction. ((64x^3 a^3/27))^{-(2/3)} = ((27/64x^3 a^3))^{(2/3)} • Step 3: Evaluate the fractional power (2/3), which means taking the cube root (denominator 3) and then squaring the result (numerator 2). Let's take the cube root of the numerator and denominator first. \sqrt[3]{27} = 3 \sqrt[3]{64x^3 a^3} = 4xa So, the expression becomes: ((3/4xa))^2 • Step 4: Square both the numerator and the denominator. (3^2/(4xa)^2) = (9/16x^2 a^2) Exam Strategy & Shortcut: To solve this rapidly, handle the constants and variables separately. For the constants: (64 \div 27)^{-2/3}. Flip it to get (27/64)^{2/3}. The cube root is 3/4, and squaring it gives 9/16. This eliminates options (a) and (b). For the variables: x^3 and a^{-3} in the denominator becomes x^3 a^3. Raising (x^3 a^3) to the power of -2/3 directly gives x^{-2} a^{-2}, which means they both belong in the denominator squared. This immediately points to option (c). Common Pitfall: A frequent error is forgetting to apply the exponent to the coefficients (64 and 27) or incorrectly flipping the fraction when dealing with the negative fractional exponent, leading to inverted answers like (16/9). Final Answer: Therefore, the correct answer is (9/16x^2 a^2).

Question 13

If $2^{n+4} - 2^{n+2} = 3$, then $n$ is equal to

Accepted Answer

Concept & Formula: The problem tests your ability to factorize exponential expressions by isolating common terms using the Laws of Indices. The fundamental rule used for factoring here is: a^{m+n} = a^m × a^n Step-by-Step Solution: • Step 1: Break down the exponents to reveal a common factor. 2^{n+4} can be written as 2^{n+2} × 2^2. Substitute this into the original equation: 2^{n+2} × 2^2 - 2^{n+2} = 3 • Step 2: Factor out the common term, which is 2^{n+2}. 2^{n+2} (2^2 - 1) = 3 • Step 3: Simplify the expression inside the parentheses. 2^{n+2} (4 - 1) = 3 2^{n+2} (3) = 3 • Step 4: Isolate the exponential term by dividing both sides by 3. 2^{n+2} = (3/3) 2^{n+2} = 1 • Step 5: Express 1 as a power of 2 to equate the bases. Any non-zero number to the power of 0 is 1, so 1 = 2^0. 2^{n+2} = 2^0 • Step 6: Equate the exponents. n + 2 = 0 n = -2 Exam Strategy & Shortcut: Instead of doing algebraic manipulation, you can use Option Elimination. Since the equation evaluates to a small integer (3), n must be a negative number or zero to keep the powers of 2 small. Try n = -2: 2^{-2+4} - 2^{-2+2} = 2^2 - 2^0 = 4 - 1 = 3. This perfectly matches the right-hand side. The answer is found in just a few seconds. Common Pitfall: A common mistake is incorrectly subtracting the bases or powers directly, mistakenly turning 2^{n+4} - 2^{n+2} into something invalid like 2^2. Remember that you can only add or subtract coefficients of identical exponential terms. Final Answer: Therefore, the correct answer is -2.

Question 14

If $2^{n-1} + 2^{n+1} = 320$, then $n$ is equal to

Accepted Answer

Concept & Formula: This question requires identifying common exponential bases and factoring them out to solve for an unknown exponent. It relies heavily on the Multiplication Law of Indices. The core principle used to break down the terms is: a^{m+n} = a^m × a^n Step-by-Step Solution: • Step 1: Express both terms on the left side in terms of the smaller power, 2^{n-1}, to facilitate easy factoring. 2^{n+1} can be rewritten as 2^{(n-1) + 2}, which is equal to 2^{n-1} × 2^2. The equation becomes: 2^{n-1} + (2^{n-1} × 2^2) = 320 • Step 2: Factor out the common 2^{n-1} term from both parts of the expression. 2^{n-1} (1 + 2^2) = 320 • Step 3: Simplify the arithmetic inside the parentheses. 2^{n-1} (1 + 4) = 320 2^{n-1} (5) = 320 • Step 4: Divide both sides by 5 to isolate the exponential term. 2^{n-1} = (320/5) 2^{n-1} = 64 • Step 5: Convert 64 into a base of 2 to match the left side. 64 = 2^6 So, 2^{n-1} = 2^6 • Step 6: Since the bases are equal, equate the exponents and solve for n. n - 1 = 6 n = 7 Exam Strategy & Shortcut: When you see 2^{n-1} + 2^{n+1}, recognize mentally that this factors to 2^{n-1}(1 + 4) = 5 × 2^{n-1}. Instantly divide 320 by 5 to get 64. Knowing your powers of 2 is crucial for aptitude exams; recognizing 64 as 2^6 means n-1 = 6, giving n = 7 almost entirely mentally. Common Pitfall: A major pitfall is improperly factoring the expression, often resulting in students forgetting the "1" when factoring out a term entirely (e.g., writing 2^{n-1}(2^2) instead of 2^{n-1}(1 + 2^2)). Always double-check by mentally distributing the factored term back. Final Answer: Therefore, the correct answer is 7.

Question 15

The value of $\frac{2^{3x+4} + 8^{x+1}}{8^{x+1} - 2^{3x+2}}$ is

Accepted Answer

Concept & Formula: This question requires converting different bases to a common prime base (2 in this case) and subsequently factoring out terms to simplify the fraction. The governing exponent rule is: (a^m)^n = a^{m × n} Step-by-Step Solution: • Step 1: Convert the base 8 terms to base 2. We know 8 = 2^3. 8^{x+1} = (2^3)^{x+1} = 2^{3(x+1)} = 2^{3x+3} • Step 2: Substitute this converted term back into the original fraction. Numerator: 2^{3x+4} + 2^{3x+3} Denominator: 2^{3x+3} - 2^{3x+2} • Step 3: Factor out the smallest power of 2 from both the numerator and the denominator. Numerator: Factor out 2^{3x+3} 2^{3x+3} (2^1 + 1) = 2^{3x+3} (3) Denominator: Factor out 2^{3x+2} 2^{3x+2} (2^1 - 1) = 2^{3x+2} (1) • Step 4: Reconstruct the fraction with the factored expressions. \frac{2^{3x+3} × 3}{2^{3x+2}} • Step 5: Simplify the exponent term using the division law. 2^{(3x+3) - (3x+2)} × 3 2^1 × 3 = 6 Exam Strategy & Shortcut: Variable Substitution: Since the options are all constants, the expression's value is independent of x. Choose the simplest value for x, which is x = 0. Substitute x = 0: Numerator: 2^4 + 8^1 = 16 + 8 = 24 Denominator: 8^1 - 2^2 = 8 - 4 = 4 Result: (24/4) = 6. This method resolves the problem in roughly 10 seconds without needing to apply index laws. Common Pitfall: A common mistake when applying the exponent rule (a^m)^n is failing to distribute the multiplier across the entire binomial. Students might write (2^3)^{x+1} as 2^{3x+1} instead of the correct 2^{3x+3}, which cascades into a completely wrong result. Final Answer: Therefore, the correct answer is 6.

Question 16

The value of $\frac{2^{n-1} - 2^n}{2^{n+4} + 2^{n+1}}$ is

Accepted Answer

Concept & Formula: The core concept is factoring exponential expressions by extracting the common term with the lowest power to simplify the numerator and denominator separately. The primary rule utilized is: a^{m+n} = a^m × a^n Step-by-Step Solution: • Step 1: Isolate the lowest power of 2 in both the numerator and the denominator to factor them out. In the numerator (2^{n-1} - 2^n), the lowest power is n-1. Note that 2^n can be written as 2^{n-1} × 2^1. In the denominator (2^{n+4} + 2^{n+1}), the lowest power is n+1. Note that 2^{n+4} can be written as 2^{n+1} × 2^3. • Step 2: Factor out these lowest power terms. Numerator: 2^{n-1} (1 - 2^1) = 2^{n-1} (1 - 2) = 2^{n-1} (-1) = -2^{n-1} Denominator: 2^{n+1} (2^3 + 1) = 2^{n+1} (8 + 1) = 2^{n+1} (9) • Step 3: Reconstruct the fraction with the simplified terms. \frac{-2^{n-1}}{9 × 2^{n+1}} • Step 4: Simplify the exponent parts by bringing 2^{n-1} down to the denominator using the division law a^m \div a^n = a^{m-n}. (-1/9 × 2^{(n+1) - (n-1))} (-1/9 × 2^{n+1-n+1)} (-1/9 × 2^2) • Step 5: Calculate the final arithmetic value. (-1/9 × 4) = -(1/36) Exam Strategy & Shortcut: Assume n=1: Since the options are all distinct numerical constants, the outcome is independent of the variable n. Choosing a small value like n=1 makes the calculation trivial. Numerator: 2^0 - 2^1 = 1 - 2 = -1 Denominator: 2^5 + 2^2 = 32 + 4 = 36 Result: -(1/36). This takes only moments to execute mentally. Common Pitfall: A very common algebraic mistake here is mishandling the subtraction in the numerator (1 - 2). Students frequently drop the negative sign by mistake and answer (1/36), which isn't an option here but could easily trap someone if it were. Final Answer: Therefore, the correct answer is -(1/36).

Question 17

If $x = 5 + 2\sqrt{6}$, then $\sqrt{x} - \frac{1}{\sqrt{x}}$ is

Accepted Answer

Concept & Formula Express a surd of the form a + 2\sqrt{b} as a perfect square (\sqrt{m} + \sqrt{n})^2 where m + n = a and m × n = b. (a+b)^2 = a^2 + b^2 + 2ab Step-by-Step Solution • Given: x = 5 + 2\sqrt{6} • We need two numbers that add to 5 and multiply to 6. These are 3 and 2. • Rewrite x: x = 3 + 2 + 2\sqrt{6} = (\sqrt{3})^2 + (\sqrt{2})^2 + 2(\sqrt{3})(\sqrt{2}) = (\sqrt{3} + \sqrt{2})^2 • Taking the square root, we get: \sqrt{x} = \sqrt{3} + \sqrt{2} • Next, find (1/\sqrt{x)} which is (1/\sqrt{3) + \sqrt{2}} • Rationalize the denominator by multiplying the numerator and denominator by the conjugate (\sqrt{3} - \sqrt{2}): (1/\sqrt{3) + \sqrt{2}} × \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} = \frac{\sqrt{3} - \sqrt{2}}{3 - 2} = \sqrt{3} - \sqrt{2} • Substitute these values into the original expression: \sqrt{x} - (1/\sqrt{x)} = (\sqrt{3} + \sqrt{2}) - (\sqrt{3} - \sqrt{2}) = \sqrt{3} + \sqrt{2} - \sqrt{3} + \sqrt{2} = 2\sqrt{2} Exam Strategy & Shortcut Whenever you have a term like \sqrt{m} + \sqrt{n} where the difference of their squares is 1 (i.e., m - n = 1), its reciprocal is simply its conjugate \sqrt{m} - \sqrt{n}. Since (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1, the reciprocal of \sqrt{3} + \sqrt{2} is instantly \sqrt{3} - \sqrt{2}. Subtracting them in your head saves critical exam time. Common Pitfall Students often make a sign error when subtracting the reciprocal. Writing -\sqrt{3} - \sqrt{2} instead of -(\sqrt{3} - \sqrt{2}) will lead to an incorrect intermediate step and a wrong option. Always use brackets when substituting terms with negative signs. Final Answer Therefore, the correct answer is 2\sqrt{2}.

Question 18

$(4 + \sqrt{7})$, expressed as a perfect square, is equal to

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Concept & Formula To convert an expression into a perfect square, we often need to introduce a 2 for the 2ab term in the algebraic identity (a+b)^2 = a^2 + b^2 + 2ab. If no 2 is present before the radical, multiply and divide the entire expression by 2. Step-by-Step Solution • Given expression: 4 + \sqrt{7} • Multiply and divide the expression by 2 to create the 2ab term: (1/2) (8 + 2\sqrt{7}) • Now, focus on the numerator 8 + 2\sqrt{7}. We need two numbers that multiply to 7 and add to 8. These are 7 and 1. • Rewrite the numerator: 8 + 2\sqrt{7} = 7 + 1 + 2\sqrt{7} = (\sqrt{7})^2 + (1)^2 + 2(\sqrt{7})(1) • This condenses into a perfect square: (\sqrt{7} + 1)^2 • Substitute this back into the fraction: (1/2) (\sqrt{7} + 1)^2 Exam Strategy & Shortcut Instead of expanding every option, you can quickly evaluate them by approximation. \sqrt{7} is approx 2.64. So 4 + 2.64 = 6.64. Option (c) is 0.5 × (2.64 + 1)^2 = 0.5 × (3.64)^2 \approx 0.5 × 13.25 = 6.62. Option (c) is the match. Alternatively, recognizing the missing 2 for the 2ab term immediately points towards multiplying/dividing by 2, which is exclusively present in Option (c). Common Pitfall A very common mistake is squaring the expression itself instead of expressing it *as* a square. Students might mistakenly square (4 + \sqrt{7}) yielding 16 + 7 + 8\sqrt{7}, which doesn't help solve the question. Final Answer Therefore, the correct answer is { (1/2)(\sqrt{7} + 1)^2 }.

Question 19

$\sqrt{8 - 2\sqrt{15}}$ is equal to

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Concept & Formula To find the square root of a surd in the form a - 2\sqrt{b}, we express the term inside the root as a perfect square (x - y)^2. (x - y)^2 = x^2 + y^2 - 2xy Step-by-Step Solution • Given: \sqrt{8 - 2\sqrt{15}} • We need to express 8 - 2\sqrt{15} as (\sqrt{m} - \sqrt{n})^2. • We need two numbers m and n such that: m + n = 8 m × n = 15 • The two numbers that satisfy both conditions are 5 and 3. • Rewrite the expression inside the square root: 8 - 2\sqrt{15} = 5 + 3 - 2\sqrt{15} = (\sqrt{5})^2 + (\sqrt{3})^2 - 2(\sqrt{5})(\sqrt{3}) • This is exactly the expansion of (\sqrt{5} - \sqrt{3})^2. • Taking the square root of the perfect square: \sqrt{(\sqrt{5} - \sqrt{3})^2} = |\sqrt{5} - \sqrt{3}| • Since \sqrt{5} > \sqrt{3}, the value is positive: \sqrt{5} - \sqrt{3}. Exam Strategy & Shortcut The fastest way to solve this is mental factoring. When you see a - 2\sqrt{b}, immediately think "factors of b that add to a." Factors of 15 adding to 8 are 5 and 3. The sign in the middle is minus, so the answer is just \sqrt{5} - \sqrt{3}. Common Pitfall A common error is putting the smaller number first, resulting in \sqrt{3} - \sqrt{5}. A principal square root must be non-negative, so you must always subtract the smaller surd from the larger surd when extracting the root. Final Answer Therefore, the correct answer is \sqrt{5} - \sqrt{3}.

Question 20

$\sqrt{6 - 4\sqrt{3} + \sqrt{16 - 8\sqrt{3}}}$ is equal to

Accepted Answer

Concept & Formula This problem involves nested square roots. The strategy is to simplify from the innermost square root outwards by turning binomial surds into perfect squares using the identity: (a - b)^2 = a^2 + b^2 - 2ab Step-by-Step Solution • Start with the innermost root: \sqrt{16 - 8\sqrt{3}} • Convert the term -8\sqrt{3} to the form -2ab. 16 - 8\sqrt{3} = 16 - 2(4\sqrt{3}) = 16 - 2\sqrt{16 × 3} = 16 - 2\sqrt{48} • We need two numbers that add to 16 and multiply to 48. These are 12 and 4. • Rewrite: (\sqrt{12})^2 + (\sqrt{4})^2 - 2\sqrt{12}\sqrt{4} = (\sqrt{12} - \sqrt{4})^2 • Simplify \sqrt{12} - \sqrt{4} to 2\sqrt{3} - 2. • So, the innermost root evaluates to 2\sqrt{3} - 2. (Note: 2\sqrt{3} > 2, so it is positive). • Substitute this back into the main expression: \sqrt{6 - 4\sqrt{3} + (2\sqrt{3} - 2)} • Combine like terms: \sqrt{4 - 2\sqrt{3}} • Now, simplify this new root. We need two numbers adding to 4 and multiplying to 3. They are 3 and 1. 4 - 2\sqrt{3} = 3 + 1 - 2\sqrt{3} = (\sqrt{3} - 1)^2 • Taking the square root gives \sqrt{3} - 1. Exam Strategy & Shortcut For nested surds, always work from the inside out. When simplifying 16 - 8\sqrt{3}, instead of pushing the whole 4 into the root to make \sqrt{48}, you can mentally break down 8\sqrt{3} as 2 × 4 × \sqrt{3}. Ask yourself: Does 4^2 + (\sqrt{3})^2 = 16? Yes! So it's immediately (4 - \sqrt{3})^2. No, wait. 16 + 3 = 19, not 16. Let's break it differently: 2 × (2\sqrt{3}) × 2. Does (2\sqrt{3})^2 + 2^2 = 16? 12 + 4 = 16. Yes! So the root is 2\sqrt{3} - 2. This mental check is much faster than full expansion. Common Pitfall When taking the square root of (x - y)^2, always ensure x > y so the result is positive. For instance, if you incorrectly write \sqrt{4 - 2\sqrt{3}} as 1 - \sqrt{3}, you will get a negative value, which violates the principle of real square roots. Final Answer Therefore, the correct answer is \sqrt{3} - 1.

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