Question 1

The value of $$ \frac{1}{(216)^{-\frac{2}{3}}} + \frac{1}{(256)^{-\frac{3}{4}}} + \frac{1}{(32)^{-\frac{1}{5}}} $$ is

Accepted Answer

Concept & Formula The problem tests the laws of negative exponents and fractional indices. a^{-m} = (1/a^m) \implies (1/a^{-m)} = a^m (x^a)^b = x^{ab} Step-by-Step Solution • First term: (1/(216)^{-\frac{2){3}}} = (216)^{(2/3)} . Since 216 = 6^3 , we apply the power rule to get (6^3)^{(2/3)} = 6^2 = 36 . • Second term: (1/(256)^{-\frac{3){4}}} = (256)^{(3/4)} . Since 256 = 4^4 , we get (4^4)^{(3/4)} = 4^3 = 64 . • Third term: (1/(32)^{-\frac{1){5}}} = (32)^{(1/5)} . Since 32 = 2^5 , we get (2^5)^{(1/5)} = 2^1 = 2 . • Add the simplified terms together: 36 + 64 + 2 = 102 . Exam Strategy & Shortcut Recognize perfect powers immediately. The numbers 216, 256, and 32 are common powers ( 6^3 , 4^4 , and 2^5 ). The denominators of the given fractional exponents (3, 4, and 5) match these powers perfectly, meaning the bases will simplify to integers. A quick mental calculation of 6^2 + 4^3 + 2^1 yields 102 in seconds without writing anything down. Common Pitfall Students often mishandle the double negative when moving a term from the denominator to the numerator, mistakenly keeping the fractional exponent negative and getting stuck with complex fractions. Remember that flipping the position vertically strictly flips the exponent's sign. Final Answer Therefore, the correct answer is 102.

Question 2

$$ (2.4 	imes 10^3) \div (8 	imes 10^{-2}) = $$ $x$

Accepted Answer

Concept & Formula This question requires combining scientific notation arithmetic and exponent rules for division. (a × 10^m/b × 10^n) = ((a/b)) × 10^{m-n} Step-by-Step Solution • Group the numerical coefficients and the base-10 powers separately: ((2.4/8)) × ((10^3/10^{-2)}) • Divide the coefficients. Since (24/8) = 3 , it follows that (2.4/8) = 0.3 . • Subtract the exponent of the denominator from the numerator's exponent: 3 - (-2) = 3 + 2 = 5 • Combine them to form 0.3 × 10^5 . • Convert to standard scientific form by moving the decimal point one place to the right, which decreases the exponent by 1: 0.3 × 10^5 = 3 × 10^4 Exam Strategy & Shortcut Treat 2.4 as 24 and divide by 8 to immediately get 3. This tells you the coefficient is related to 3. Then, quickly calculate the power of 10: 3 - (-2) = 5 . Since we shifted a decimal (0.3 instead of 3), 10^5 becomes 10^4 . You can mentally verify the order of magnitude to lock in the correct option in under 15 seconds. Common Pitfall Students often mishandle the subtraction of a negative exponent, mistakenly calculating 3 - 2 = 1 instead of 3 - (-2) = 5 . Always write out or mentally vocalize the double negative explicitly when dividing exponents with negative signs. Final Answer Therefore, the correct answer is 3 × 10^4 .

Question 3

$$ \left(\frac{1}{216}\right)^{-\frac{2}{3}} \div \left(\frac{1}{27}\right)^{-\frac{4}{3}} = $$ $x$

Accepted Answer

Concept & Formula This problem tests the inversion rule for negative exponents applied to fractions, followed by exponent simplification. ((a/b))^{-n} = ((b/a))^n (x^a)^b = x^{ab} Step-by-Step Solution • First, eliminate the negative signs in the exponents by reciprocating the inner fractions: ((1/216))^{-(2/3)} = (216)^{(2/3)} ((1/27))^{-(4/3)} = (27)^{(4/3)} • Convert the bases into their prime power equivalents. 216 = 6^3 and 27 = 3^3 . • Apply the power of a power rule: (6^3)^{(2/3)} = 6^2 = 36 (3^3)^{(4/3)} = 3^4 = 81 • Substitute these simplified values back into the original division operation: 36 \div 81 = (36/81) • Simplify the fraction by dividing the numerator and denominator by their greatest common divisor (9): (36 \div 9/81 \div 9) = (4/9) Exam Strategy & Shortcut Flip the fractions immediately to remove negative powers. Recognize that 216 and 27 are cubes. The denominator 3 in the fractional exponents signals that taking the cube root is your first step. Cube root of 216 is 6; square it to get 36. Cube root of 27 is 3; raise it to the 4th power to get 81. The ratio 36/81 visually reduces to 4/9 quickly using the 9 times table. Common Pitfall A major trap is attempting to perform the division operation before simplifying the individual exponential terms. This leads to convoluted algebraic expressions like (1/216 \div 1/27) , which jumbles the differing exponents and nearly guarantees a calculation error. Always simplify bases independently first. Final Answer Therefore, the correct answer is (4/9) .

Question 4

$$ (48)^{-\frac{2}{7}} 	imes (16)^{-\frac{5}{7}} 	imes (3)^{-\frac{5}{7}} = $$ $x$

Accepted Answer

Concept & Formula The problem tests the properties of exponents, specifically combining bases with identical powers. a^n × b^n = (ab)^n a^m × a^n = a^{m+n} Step-by-Step Solution • Notice that the last two terms share the exact same fractional exponent: -(5/7) . • Group these terms using the rule a^n × b^n = (ab)^n : (16)^{-(5/7)} × (3)^{-(5/7)} = (16 × 3)^{-(5/7)} = (48)^{-(5/7)} • Now, substitute this simplified combined term back into the original expression: (48)^{-(2/7)} × (48)^{-(5/7)} • Since the bases are now identical (48), simply add the exponents: -(2/7) + (-(5/7)) = -(7/7) = -1 • The expression simplifies to (48)^{-1} . • Apply the negative exponent rule a^{-1} = (1/a) to yield (1/48) . Exam Strategy & Shortcut Always scan for relationships between numbers before blindly factoring into primes. Spotting that 16 × 3 = 48 instantly collapses the complicated terms into a single base (48), turning a 2-minute prime factorization slog into a 10-second mental addition of fractions. Look for synergy in the bases when exponents match! Common Pitfall A common mistake is completely breaking down 48, 16, and 3 into absolute prime factors (powers of 2 and 3) right at the start. While this method mathematically works, it is heavily prone to sign and arithmetic errors with the numerous fractional exponents and wastes valuable exam time. Final Answer Therefore, the correct answer is (1/48) .

Question 5

If $$ 10^x = \frac{1}{2} $$, then $$ 10^{-8x} = $$ $x$

Accepted Answer

Concept & Formula This problem is solved using the "Power of a Power" rule in exponents, which allows us to manipulate the given equation to match the target expression. (a^m)^n = a^{m × n} a^{-m} = (1/a^m) Step-by-Step Solution • We are given that 10^x = (1/2) . • We need to find the value of 10^{-8x} . We can rewrite this target expression using the power of a power rule: 10^{-8x} = (10^x)^{-8} • Now, substitute the given value of 10^x into our rewritten expression: ((1/2))^{-8} • Apply the negative exponent rule to flip the fraction and make the exponent positive: ((2/1))^8 = 2^8 • Calculate the final value: 2^8 = 256 Exam Strategy & Shortcut Instead of trying to solve for x using logarithms (which wastes massive amounts of time), manipulate the base structure. You need -8x in the exponent, so immediately raise both sides of the known equation to the power of -8. The right side becomes (1/2)^{-8}. Since flipping the fraction negates the exponent, this is simply 2^8. Memorizing powers of 2 up to 2^{10} (1024) makes calculating 2^8 = 256 an instantaneous mental step. Common Pitfall A frequent trap is ignoring the negative sign in the exponent and simply calculating (1/2)^8 = 1/256, leading students straight to option (a). Always remember that a negative exponent on a fraction effectively turns it upside down. Final Answer Therefore, the correct answer is 256.

Question 6

If $$ \left(\frac{3}{5}\right)^3 \left(\frac{3}{5}\right)^{-6} = \left(\frac{3}{5}\right)^{2x-1} $$, then $x$ is equal to

Accepted Answer

Concept & Formula This problem tests the product rule of exponents and the property of equating exponents when the bases on both sides of an equation are identical. a^m × a^n = a^{m+n} If a^m = a^n , then m = n (given a 
eq 0, 1, -1 ). Step-by-Step Solution • Observe the left-hand side of the equation. Both terms have the same base: (3/5) . • Apply the product rule by adding the exponents: ((3/5))^{3 + (-6)} = ((3/5))^{-3} • Now, rewrite the full equation with the simplified left side: ((3/5))^{-3} = ((3/5))^{2x-1} • Since the bases are exactly the same on both sides, we can drop the bases and equate the exponents directly: -3 = 2x - 1 • Solve the linear equation for x. Add 1 to both sides: -3 + 1 = 2x -2 = 2x • Divide by 2: x = -1 Exam Strategy & Shortcut Since the bases across the entire equation are identical (3/5), you can bypass writing them down entirely. Jump straight to the exponent arithmetic: 3 - 6 = 2x - 1. That simplifies mentally to -3 = 2x - 1, meaning 2x = -2, so x = -1. This takes less than 5 seconds without writing a single fraction. Common Pitfall A common error is mishandling the negative numbers during basic addition and subtraction. Students sometimes calculate 3 - 6 = 3 instead of -3, which leads to the incorrect equation 3 = 2x - 1, resulting in x = 2. Pay strict attention to signs during exponent arithmetic. Final Answer Therefore, the correct answer is -1.

Question 7

$$ 49 	imes 49 	imes 49 	imes 49 = 7^x $$

Accepted Answer

Concept & Formula This question evaluates the ability to express composite numbers using prime bases and applying the exponent rules for multiplication. (a^m)^n = a^{m × n} a^m × a^n = a^{m+n} Step-by-Step Solution • The right side of the equation uses a base of 7. To equate both sides, convert the numbers on the left side to base 7. • We know that 49 = 7^2 . • Rewrite the original expression by replacing every 49 with 7^2 : 7^2 × 7^2 × 7^2 × 7^2 = 7^x • Apply the product rule for exponents. Since the bases are identical, add all the exponents together: 2 + 2 + 2 + 2 = 8 • This simplifies the left side to 7^8 . • Equate both sides: 7^8 = 7^x • Therefore, equating the exponents gives x = 8 . Exam Strategy & Shortcut Recognize the relationship between 49 and 7 immediately. Since there are four instances of 49 being multiplied, and each 49 is a "square" (7^2), simply multiply the number of terms (4) by the power (2). 4 × 2 = 8. You arrive at the correct exponent without writing down the intermediate steps. Common Pitfall A frequent mistake is to just count the number of times 49 appears (which is 4) and blindly select option (a). Remember that the base changes from 49 to 7, so you must account for the square factor. 49^4 is not equal to 7^4. Final Answer Therefore, the correct answer is 8.

Question 8

The value of $$ (8^{-25} - 8^{-26}) $$ is

Accepted Answer

Concept & Logic The problem tests your ability to factor out common terms with negative exponents. When dealing with negative exponents, the term with the larger absolute value in the exponent is actually the smaller quantity. We factor out the smaller term. a^{-m} - a^{-(m+1)} = a^{-(m+1)} (a^1 - 1) Step-by-Step Solution • Identify the smaller term. Between 8^{-25} and 8^{-26} , 8^{-26} is the smaller value. • Rewrite the larger term ( 8^{-25} ) in terms of the smaller term: 8^{-25} = 8^{-26} × 8^1 • Substitute this back into the original expression: (8^{-26} × 8^1) - 8^{-26} • Factor out the common term, 8^{-26} : 8^{-26}(8^1 - 1) • Simplify the expression inside the parentheses: 8 - 1 = 7 • Combine the terms for the final factored form: 7 × 8^{-26} Exam Strategy & Shortcut When subtracting powers of the same base that are exactly 1 unit apart (like -25 and -26), you can instantly apply a shortcut rule: factor out the smaller power and multiply by (base - 1). Here, the base is 8. The smaller power is 8^{-26} . Multiply by (8 - 1) = 7 . The answer is instantly 7 × 8^{-26} . This takes 5 seconds mentally. Common Pitfall A very common mistake is factoring out 8^{-25} instead of 8^{-26} , which leaves a negative exponent inside the parenthesis, creating unnecessary fractions and confusion. Always factor out the lowest mathematical value, which is the "most negative" exponent. Final Answer Therefore, the correct answer is 7 × 8^{-26} .

Question 9

$$ (64)^{-\frac{1}{2}} - (-32)^{-\frac{4}{5}} = $$ $x$

Accepted Answer

Concept & Formula This question combines the evaluation of fractional exponents with the application of negative bases. (x^m)^{(1/n)} = (x^{(1/n)})^m a^{-n} = (1/a^n) Step-by-Step Solution • Evaluate the first term: (64)^{-(1/2)} Since 64 = 8^2 , apply the power rule: (8^2)^{-(1/2)} = 8^{-1} = (1/8) • Evaluate the second term: (-32)^{-(4/5)} Recognize that -32 = (-2)^5 . Apply the power rule: ((-2)^5)^{-(4/5)} = (-2)^{-4} • Convert the negative exponent into a fraction: (-2)^{-4} = (1/(-2)^4) Since any negative number raised to an even power becomes positive: (-2)^4 = 16 So, the second term simplifies to (1/16) . • Substitute both simplified terms back into the main expression: (1/8) - (1/16) • Find a common denominator (16) and subtract: (2/16) - (1/16) = (1/16) Exam Strategy & Shortcut Break down complex terms independently. For the second term, immediately note the denominator of the exponent is 5, meaning you need a 5th root. The 5th root of -32 is exactly -2. Then, raise -2 to the remaining numerator power, which is -4. So, it becomes (-2)^{-4}, yielding 1/16. The first term is simply the reciprocal of the square root of 64, giving 1/8. Simply calculate 1/8 - 1/16 = 1/16 instantly. Common Pitfall The most dangerous pitfall here is mishandling the negative base in the second term. Students often see the negative sign in -32 and incorrectly carry it forward to make the final fraction negative, completely forgetting that raising a negative number to an even power (the 4 in the exponent numerator) makes the result positive. Final Answer Therefore, the correct answer is (1/16) .

Question 10

If $$ \left(\frac{a}{b}\right)^{x-1} = \left(\frac{b}{a}\right)^{x-3} $$, then the value of $x$ is

Accepted Answer

Concept & Formula This problem tests the inversion rule for exponents, which allows us to manipulate fractions to match their bases on both sides of an equation. ((a/b))^{-n} = ((b/a))^n If m^x = m^y , then x = y . Step-by-Step Solution • The given equation has bases that are reciprocals of each other: (a/b) and (b/a) . • We need to make the bases identical. Let's invert the base on the right side of the equation. • According to the inversion rule, flipping the fraction means we must multiply the entire exponent by -1: ((b/a))^{x-3} = ((a/b))^{-(x-3)} = ((a/b))^{-x+3} • Now, rewrite the original equation using this new equivalent term: ((a/b))^{x-1} = ((a/b))^{-x+3} • Since the bases are now exactly the same ( (a/b) ), we can equate their exponents directly: x - 1 = -x + 3 • Solve the resulting linear equation for x. Add x to both sides: 2x - 1 = 3 • Add 1 to both sides: 2x = 4 • Divide by 2: x = 2 Exam Strategy & Shortcut Whenever you see an equation with fractional bases flipped (like a/b and b/a), immediately negate one of the exponents and set them equal to each other. You don't need to write out the base change steps. Simply write: (x - 1) = -(x - 3) . This expands to x - 1 = -x + 3 , leading to 2x = 4 , and finally x = 2 . This reduces a 1-minute algebraic process into a 10-second mental calculation. Common Pitfall A common error is failing to distribute the negative sign across the entire binomial exponent when inverting the fraction. Students often incorrectly rewrite the right-side exponent as -x - 3 instead of -x + 3, which fundamentally ruins the final equation and leads to incorrect answers. Always use parentheses when negating. Final Answer Therefore, the correct answer is 2.

Question 11

If $5\sqrt{5} 	imes 5^3 \div 5^{-\frac{3}{2}} = 5^{a+2}$, then the value of $a$ is

Accepted Answer

Concept & Formula: This problem tests the application of multiple laws of indices simultaneously. Convert all radicals into fractional exponents, and combine terms with the same base using addition for multiplication and subtraction for division. Key rules applied: x^m × x^n = x^{m+n} x^m \div x^n = x^{m-n} \sqrt{x} = x^{(1/2)} Step-by-Step Solution: • Given: 5\sqrt{5} × 5^3 \div 5^{-(3/2)} = 5^{a+2} • Calculation: Convert the surd into an exponent: 5\sqrt{5} = 5^1 × 5^{(1/2)} = 5^{1 + 0.5} = 5^{1.5} Rewrite the entire equation with explicit exponents: 5^{1.5} × 5^3 \div 5^{-1.5} = 5^{a+2} Apply the product rule (add exponents for multiplication): 5^{1.5 + 3} \div 5^{-1.5} = 5^{a+2} 5^{4.5} \div 5^{-1.5} = 5^{a+2} Apply the quotient rule (subtract exponents for division): 5^{4.5 - (-1.5)} = 5^{a+2} 5^{4.5 + 1.5} = 5^{a+2} 5^6 = 5^{a+2} Equate the exponents since the bases are identical: 6 = a + 2 a = 4 Exam Strategy & Shortcut: Work Only with Exponents: Once you verify the base is uniformly 5, completely drop the base and write a linear equation of only the exponents to save time: 1 + 0.5 + 3 - (-1.5) = a + 2 4.5 + 1.5 = a + 2 6 = a + 2 \Rightarrow a = 4 This takes less than 10 seconds. Common Pitfall: The most common error is mishandling the division of a negative exponent. Students often write 4.5 - 1.5 = 3 instead of correctly subtracting the negative value 4.5 - (-1.5) = 6. Always pay strict attention to double negatives. Final Answer: Therefore, the correct answer is 4.

Question 12

If $\sqrt{2^n} = 64$, then the value of $n$ is

Accepted Answer

Concept & Formula: This question requires converting a square root expression into a fractional exponent and mapping a whole number to its prime base power to equate the exponents. The critical translations are: \sqrt{x} = x^{(1/2)} (\sqrt{x})^2 = x Step-by-Step Solution: • Given: \sqrt{2^n} = 64 • Calculation (Method 1: Fractional Exponents): Express the square root as a power of (1/2): (2^n)^{(1/2)} = 64 2^{(n/2)} = 64 Convert 64 into a base of 2: 64 = 2^6 Substitute and equate the bases: 2^{(n/2)} = 2^6 (n/2) = 6 n = 12 Exam Strategy & Shortcut: Squaring Both Sides: An extremely fast way to bypass fractional exponents is to immediately square both sides of the original equation to eliminate the radical. (\sqrt{2^n})^2 = 64^2 2^n = (2^6)^2 2^n = 2^{12} n = 12 Common Pitfall: A very frequent mistake is identifying 64 as 2^6 and then immediately assuming n = 6, entirely ignoring the square root over the 2^n term. Always isolate the variable entirely. Final Answer: Therefore, the correct answer is 12.

Question 13

If $(\sqrt{3})^5 	imes 9^2 = 3^n 	imes 3\sqrt{3}$, then the value of $n$ is

Accepted Answer

Concept & Formula: The core concept is to convert all bases and surds (roots) into a common prime base, which in this case is 3. By expressing all terms as powers of 3, we can use the laws of indices to combine terms and equate the exponents on both sides of the equation. The necessary laws of indices are: \sqrt{x} = x^{(1/2)} x^m × x^n = x^{m+n} (x^m)^n = x^{mn} Step-by-Step Solution: • Given: (\sqrt{3})^5 × 9^2 = 3^n × 3\sqrt{3} • Calculation: Convert all terms to base 3. For the left-hand side (LHS): \sqrt{3} = 3^{(1/2)}, so (\sqrt{3})^5 = (3^{(1/2)})^5 = 3^{(5/2)} = 3^{2.5} 9 = 3^2, so 9^2 = (3^2)^2 = 3^4 Multiply these together using the product rule (add the exponents): LHS = 3^{2.5} × 3^4 = 3^{2.5 + 4} = 3^{6.5} For the right-hand side (RHS): 3\sqrt{3} = 3^1 × 3^{(1/2)} = 3^{1 + 0.5} = 3^{1.5} Multiply this with 3^n: RHS = 3^n × 3^{1.5} = 3^{n + 1.5} Equate the simplified LHS and RHS: 3^{6.5} = 3^{n + 1.5} Since the bases are identical, equate the exponents: 6.5 = n + 1.5 n = 6.5 - 1.5 n = 5 Exam Strategy & Shortcut: Work Exclusively with Exponents: Once you identify the common base is 3, skip writing the base and only write the arithmetic for the exponents. LHS Exponents: 5 × 0.5 + 2 × 2 = 2.5 + 4 = 6.5 RHS Exponents: n + 1 + 0.5 = n + 1.5 Set them equal: 6.5 = n + 1.5 \Rightarrow n = 5. This can be done entirely mentally in 15 seconds. Common Pitfall: Students often misinterpret 3\sqrt{3}. They might accidentally treat it as an addition or miscalculate the combined power as just 3^{(1/2)} instead of recognizing it is 3^1 × 3^{(1/2)} = 3^{(3/2)}. Final Answer: Therefore, the correct answer is 5.

Question 14

If $\left(\frac{9}{4}\right)^x \cdot \left(\frac{8}{27}\right)^{x-1} = \frac{2}{3}$, then the value of $x$ is

Accepted Answer

Concept & Formula: When an equation features fractions as bases, the objective is to invert or manipulate them until they share identical fractional bases. This relies heavily on recognizing perfect squares and cubes of small fractions, and using negative exponents to flip fractions. The pivotal rules are: ((a/b))^{-n} = ((b/a))^n (x^m)^n = x^{mn} Step-by-Step Solution: • Given: ((9/4))^x \cdot ((8/27))^{x-1} = (2/3) • Calculation: Recognize that the fractions are related to base (3/2) and (2/3). Let us standardize to base (3/2). Convert the first term: (9/4) = ((3/2))^2 So, ((9/4))^x = (((3/2))^2)^x = ((3/2))^{2x} Convert the second term: (8/27) = ((2/3))^3 To match our base of (3/2), flip it and negate the exponent: ((2/3))^3 = ((3/2))^{-3} So, ((8/27))^{x-1} = (((3/2))^{-3})^{x-1} = ((3/2))^{-3(x-1)} = ((3/2))^{-3x+3} Convert the right-hand side: (2/3) = ((3/2))^{-1} Substitute everything back into the equation: ((3/2))^{2x} \cdot ((3/2))^{-3x+3} = ((3/2))^{-1} Combine the LHS exponents using addition: ((3/2))^{2x - 3x + 3} = ((3/2))^{-1} ((3/2))^{-x + 3} = ((3/2))^{-1} Equate the exponents: -x + 3 = -1 -x = -4 x = 4 Exam Strategy & Shortcut: Option Verification: For fractional equations that look algebraically messy, plugging in the options can be faster. Try Option (d) where x = 4: LHS = ((9/4))^4 \cdot ((8/27))^3 = (3^8/2^8) \cdot (2^9/3^9) = (3^8 \cdot 2^9/2^8 \cdot 3^9) = (2^1/3^1) = (2/3) This perfectly matches the RHS, confirming the answer instantly without full algebra. Common Pitfall: Failing to distribute the negative sign properly when multiplying the exponents is the most frequent error. Writing -3(x-1) as -3x-3 instead of the correct -3x+3 leads to a completely incorrect final value for x. Final Answer: Therefore, the correct answer is 4.

Question 15

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 16

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 17

$\frac{2^{n+4} - 2(2^n)}{2(2^{n+3})}$ 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 18

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

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 19

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

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 20

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

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.

Surds and Indices Questions

Questions

Practice smarter