Question 1

Given $2^x = 8^{y+1}$ and $9^y = 3^{x-9}$, the value of $x+y$ is

Accepted Answer

Concept & Formula: This problem requires converting separate exponential equations into a system of linear equations by matching their prime bases. Rule: If a^m = a^n, then m = n (given a > 0 and a 
eq 1). Step-by-Step Solution: • Step 1: Simplify the first equation by writing both sides with a base of 2. Given: 2^x = 8^{y+1} Since 8 = 2^3 : 2^x = (2^3)^{y+1} 2^x = 2^{3y+3} Equating the exponents gives our first linear equation: (1) x = 3y + 3 • Step 2: Simplify the second equation by writing both sides with a base of 3. Given: 9^y = 3^{x-9} Since 9 = 3^2 : (3^2)^y = 3^{x-9} 3^{2y} = 3^{x-9} Equating the exponents gives our second linear equation: (2) 2y = x - 9 • Step 3: Solve the linear system using substitution. Substitute equation (1) into equation (2): 2y = (3y + 3) - 9 2y = 3y - 6 • Step 4: Isolate y. 6 = 3y - 2y y = 6 • Step 5: Substitute y=6 back into equation (1) to find x. x = 3(6) + 3 x = 18 + 3 = 21 • Step 6: Calculate the final required value, x+y. x + y = 21 + 6 = 27 Exam Strategy & Shortcut: Reverse Engineering from Options: If you derive x = 3y+3 quickly, notice that x is approximately 3 times y. So x+y is roughly 4y. Since 4y \approx Total, test y-values that yield integer x values. Try y=5 \implies x=18 \implies Sum=23 (not an option). Try y=6 \implies x=21 \implies Sum=27 (Option d). Verify with the second equation: 2(6) = 21 - 9 \implies 12 = 12. This confirms 27 is the correct answer rapidly without full algebraic substitution. Common Pitfall: The most frequent error is calculating the values of x (21) or y (6) and instinctively looking for those numbers in the options, forgetting that the question specifically asks for the sum x+y. Always re-read the final sentence of the prompt before selecting an answer. Final Answer: Therefore, the correct answer is 27.

Question 2

What are the values of $x$ and $y$ that satisfy the equation $2^{0.7x} \cdot 3^{-1.25y} = \frac{8\sqrt{6}}{27}$?

Accepted Answer

Concept & Strategy: The key to solving equations with unknown variables in the exponents of different prime bases is to prime factorize both sides of the equation. By expressing all terms as powers of their distinct prime bases (here, 2 and 3), you can then equate the exponents of corresponding bases on both sides. a^m \cdot b^n = a^p \cdot b^q \implies m = p and n = q Step-by-Step Solution: • Given: 2^{0.7x} \cdot 3^{-1.25y} = \frac{8\sqrt{6}}{27} • Calculation / Deduction: 1. First, break down the Right Hand Side (RHS) into prime bases of 2 and 3. We know that 8 = 2^3, 27 = 3^3, and \sqrt{6} = \sqrt{2 \cdot 3} = 2^{0.5} \cdot 3^{0.5}. 2. Substitute these back into the RHS: RHS = \frac{2^3 \cdot (2^{0.5} \cdot 3^{0.5})}{3^3} 3. Combine the powers in the numerator and bring the denominator up by changing the sign of its exponent: RHS = 2^{3 + 0.5} \cdot 3^{0.5 - 3} RHS = 2^{3.5} \cdot 3^{-2.5} 4. Now, equate the Left Hand Side (LHS) to the simplified RHS: 2^{0.7x} \cdot 3^{-1.25y} = 2^{3.5} \cdot 3^{-2.5} 5. Equate the exponents of base 2: 0.7x = 3.5 \implies x = (3.5/0.7) = 5 6. Equate the exponents of base 3: -1.25y = -2.5 \implies y = (-2.5/-1.25) = 2 Exam Strategy & Shortcut: Option Elimination: Once you simplify the RHS to 2^{3.5} \cdot 3^{-2.5}, you immediately know that 0.7x = 3.5. This means x = 5. Looking at the options, only option (d) has x = 5. You can confidently mark (d) without even calculating for y, saving precious seconds. Common Pitfall: A common mistake is incorrectly expanding \sqrt{6}. Students sometimes forget that the square root applies to both prime factors, leading to an incorrect power for either 2 or 3. Always remember \sqrt{ab} = a^{0.5} \cdot b^{0.5}. Final Answer: Therefore, the correct answer is x = 5, y = 2.

Question 3

Let $r$ be the result of doubling both the base and the exponent of $a^b$, $b 
eq 0$. If $r$ equals the product of $a^b$ by $x^b$, then $x$ equals

Accepted Answer

Concept & Formula: This problem tests the fundamental rules of exponents, specifically the power of a product rule and how to equate expressions with the same exponent. The core principle is recognizing how changes to the base and exponent affect the algebraic expression. (xy)^n = x^n \cdot y^n Step-by-Step Solution: • Given: The original term is a^b. r is the result of doubling the base (from a to 2a) and the exponent (from b to 2b). Also, r = a^b \cdot x^b. • Calculation / Deduction: 1. Express r algebraically based on the first condition: r = (2a)^{2b} 2. Express the second condition mathematically: r = a^b \cdot x^b 3. Combine the two equations: (2a)^{2b} = a^b \cdot x^b 4. Simplify the RHS using the rule (xy)^n = x^n \cdot y^n: (2a)^{2b} = (ax)^b 5. To compare bases, we need the exponents to match. Rewrite the LHS so it has an exponent of b: ((2a)^2)^b = (ax)^b 6. Expand the inner term on the LHS: (4a^2)^b = (ax)^b 7. Since the exponents are identical and non-zero (b 
eq 0), we can equate the bases: 4a^2 = ax 8. Solve for x: x = (4a^2/a) = 4a Exam Strategy & Shortcut: Value Assumption (Plug-in Method): Let a = 2 and b = 1. Original term = 2^1 = 2. Double base (2 ightarrow 4) and exponent (1 ightarrow 2). So, r = 4^2 = 16. Given r = a^b \cdot x^b \implies 16 = 2^1 \cdot x^1 \implies 2x = 16 \implies x = 8. Now check the options with a = 2: (a) 2 (b) 4 (c) 2a = 2(2) = 4 (d) 4a = 4(2) = 8 Option (d) yields 8, perfectly matching our result. Common Pitfall: The most frequent error is misinterpreting "doubling both the base and the exponent". Students often write 2(a^b) or (2a)^b instead of properly squaring the entire new base and doubling the power to yield (2a)^{2b}. Final Answer: Therefore, the correct answer is 4a.

Question 4

Which of the following is the greatest?

Accepted Answer

Concept & Strategy: To compare different surds (roots) of different numbers, you cannot easily judge them in their radical form. The most reliable strategy is to convert them to fractional exponents and then raise all the terms to a common power. This common power should be the Least Common Multiple (LCM) of the denominators of the fractional exponents. x^{(1/a)} and y^{(1/b)} \rightarrow Raise to LCM(a,b) Step-by-Step Solution: • Given: The numbers are \sqrt{2}, \sqrt[3]{3}, \sqrt[4]{4}, \sqrt[6]{6}. • Calculation / Deduction: 1. Convert the radicals into fractional exponents: 2^{(1/2)}, 3^{(1/3)}, 4^{(1/4)}, 6^{(1/6)} 2. Identify the denominators of the exponents: 2, 3, 4, and 6. 3. Find the LCM of these denominators: LCM(2, 3, 4, 6) = 12 4. Raise each term to the power of 12 to eliminate the fractions, making them easy to compare: • (2^{(1/2)})^{12} = 2^{12/2} = 2^6 = 64 • (3^{(1/3)})^{12} = 3^{12/3} = 3^4 = 81 • (4^{(1/4)})^{12} = 4^{12/4} = 4^3 = 64 • (6^{(1/6)})^{12} = 6^{12/6} = 6^2 = 36 5. Compare the resulting integers: 81 > 64 = 64 > 36. 6. The greatest value is 81, which corresponds to the original term \sqrt[3]{3}. Exam Strategy & Shortcut: Quick Simplification Check: Notice immediately that \sqrt[4]{4} = (2^2)^{(1/4)} = 2^{(2/4)} = 2^{(1/2)} = \sqrt{2}. Since options (a) and (c) are mathematically identical, neither can be the unique "greatest" answer in a standard MCQ. This immediately narrows your choices down to just comparing \sqrt[3]{3} and \sqrt[6]{6}. Raise both to the power of 6: (\sqrt[3]{3})^6 = 3^2 = 9 (\sqrt[6]{6})^6 = 6^1 = 6 Since 9 > 6, \sqrt[3]{3} is clearly the greatest. Common Pitfall: Students often attempt to approximate decimal values for these roots without calculating them properly, leading to guesses. Without memorizing \sqrt[3]{3} \approx 1.442, guessing based on the base numbers (thinking 6 is the largest base so it must be greatest) is a very common trap. Final Answer: Therefore, the correct answer is \sqrt[3]{3}.

Question 5

The greatest of $\sqrt{2}, \sqrt[6]{3}, \sqrt[3]{4}, \sqrt[4]{5}$ is

Accepted Answer

Concept & Strategy: Just like other root comparison problems, the core strategy here is to eliminate the radical signs. We achieve this by converting roots into fractional exponents and then multiplying all exponents by the Least Common Multiple (LCM) of their denominators. This scales the numbers proportionally, allowing for a straightforward integer comparison. Compare a^{(1/x)}, b^{(1/y)} \implies Compare a^{\frac{LCM}{x}}, b^{\frac{LCM}{y}} Step-by-Step Solution: • Given: The values to compare are \sqrt{2}, \sqrt[6]{3}, \sqrt[3]{4}, \sqrt[4]{5}. • Calculation / Deduction: 1. Express each root as a base with a fractional exponent: 2^{(1/2)}, 3^{(1/6)}, 4^{(1/3)}, 5^{(1/4)} 2. Extract the denominators from the exponents: 2, 6, 3, and 4. 3. Calculate the LCM of (2, 6, 3, 4): LCM(2, 6, 3, 4) = 12 4. Raise every term to the power of 12 to transform them into whole numbers: • (\sqrt{2})^{12} = (2^{(1/2)})^{12} = 2^6 = 64 • (\sqrt[6]{3})^{12} = (3^{(1/6)})^{12} = 3^2 = 9 • (\sqrt[3]{4})^{12} = (4^{(1/3)})^{12} = 4^4 = 256 • (\sqrt[4]{5})^{12} = (5^{(1/4)})^{12} = 5^3 = 125 5. Compare the simplified integer values: 256 > 125 > 64 > 9. 6. The highest value is 256, which maps back to the original term \sqrt[3]{4}. Exam Strategy & Shortcut: Rapid Elimination: You can compare them in pairs to save mental load. Compare \sqrt{2} and \sqrt[3]{4}: Raise both to power 6. 2^3 = 8 vs 4^2 = 16. So \sqrt[3]{4} is greater. Compare \sqrt[6]{3} and \sqrt[4]{5}: Raise both to power 12. 3^2 = 9 vs 5^3 = 125. So \sqrt[4]{5} is greater. Now just compare the winners: \sqrt[3]{4} and \sqrt[4]{5}. Raise both to power 12. 4^4 = 256 vs 5^3 = 125. \sqrt[3]{4} is the definitive winner. Breaking it down this way avoids juggling 4 large calculations at once. Common Pitfall: A common trap is mixing up the order of the original roots when checking the options. Notice the question lists them in one order (\sqrt{2}, \sqrt[6]{3}, \sqrt[3]{4}, \sqrt[4]{5}) but the options shuffle them. Always explicitly map your final highest integer back to its exact initial radical form before selecting an option. Final Answer: Therefore, the correct answer is \sqrt[3]{4}.

Question 6

The largest number in the sequence $1, 2^{\frac{1}{2}}, 3^{\frac{1}{3}}, 4^{\frac{1}{4}}, \ldots, n^{\frac{1}{n}}$ is

Accepted Answer

Concept & Strategy: This question touches upon a classic calculus optimization problem disguised as a sequence. The sequence represents the function f(x) = x^{(1/x)}. To find the maximum value of this sequence for integer values of x, we evaluate the function around its global maximum peak, which occurs at x = e (Euler's number, approximately 2.718). Since e falls between the integers 2 and 3, the maximum value in this integer sequence must occur at either n = 2 or n = 3. Step-by-Step Solution: • Given: The sequence n^{(1/n)} for natural numbers n = 1, 2, 3, 4 \ldots • Calculation / Deduction: 1. By mathematical theory, the sequence increases until it hits e and strictly decreases thereafter. Therefore, we only need to compare the integer values adjacent to e (which are 2 and 3). 2. Let's manually compare n=2 and n=3: We need to determine which is larger: 2^{(1/2)} or 3^{(1/3)}. 3. Take the LCM of the denominators of the powers (2 and 3), which is 6. 4. Raise both terms to the 6th power to remove fractional exponents: (2^{(1/2)})^6 = 2^3 = 8 (3^{(1/3)})^6 = 3^2 = 9 5. Since 9 > 8, we know that 3^{(1/3)} > 2^{(1/2)}. 6. Because the sequence strictly decreases for all integers n \geq 3, 3^{(1/3)} must be the absolute largest number in the entire sequence. Exam Strategy & Shortcut: Fact Memorization: For competitive exams like CAT/IIFT, the function y = x^{(1/x)} is a standard curve to memorize. Its absolute maximum always occurs at x = e \approx 2.718. Since we are dealing with a sequence of integers, the maximum must be the integer closest to 2.718, which intuitively points to 3. A quick check of 2 vs 3 confirms 3^{1/3} is indeed the peak. Common Pitfall: Many students mistakenly believe that as the base n increases, the overall value of n^{(1/n)} must also continuously increase. They might guess options (c) or (d) thinking larger numbers yield larger results. Always remember that a growing denominator in an exponent (1/n) shrinks the value drastically, overpowering the growing base. Final Answer: Therefore, the correct answer is 3^{(1/3)}.

Question 7

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

Accepted Answer

Concept & Formula: The key to simplifying expressions with surds under square roots is to express the inner surd as a perfect square. This is done by splitting the integer part into two numbers that add up to it, and whose product relates to the surd part. (a + b)^2 = a^2 + b^2 + 2ab Step-by-Step Solution: • Given: x = 5 + 2\sqrt{6} Expression to evaluate: (x - 1/\sqrt{x)} • Calculation / Deduction: 1. First, simplify the given expression by separating the terms: (x - 1/\sqrt{x)} = (x/\sqrt{x)} - (1/\sqrt{x)} = \sqrt{x} - (1/\sqrt{x)} 2. Next, find the square root of x. We need to express 5 + 2\sqrt{6} in the form of (a + b)^2. 3. Notice that 5 = 3 + 2 and \sqrt{6} = \sqrt{3 × 2}. x = 3 + 2 + 2\sqrt{3}\sqrt{2} = (\sqrt{3})^2 + (\sqrt{2})^2 + 2(\sqrt{3})(\sqrt{2}) x = (\sqrt{3} + \sqrt{2})^2 4. Taking the square root of both sides: \sqrt{x} = \sqrt{3} + \sqrt{2} 5. Now find its reciprocal, (1/\sqrt{x)}. Rationalize the denominator: (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} 6. Substitute these back into our simplified expression: \sqrt{x} - (1/\sqrt{x)} = (\sqrt{3} + \sqrt{2}) - (\sqrt{3} - \sqrt{2}) \sqrt{x} - (1/\sqrt{x)} = \sqrt{3} + \sqrt{2} - \sqrt{3} + \sqrt{2} = 2\sqrt{2} Exam Strategy & Shortcut: Algebraic Manipulation: Instead of finding \sqrt{x} first, you can manipulate the final required expression. If y = \sqrt{x} - (1/\sqrt{x)}, then squaring both sides gives y^2 = x + (1/x) - 2. Given x = 5 + 2\sqrt{6}, the reciprocal is (1/x) = 5 - 2\sqrt{6} (since 5^2 - (2\sqrt{6})^2 = 25 - 24 = 1). So, x + (1/x) = (5 + 2\sqrt{6}) + (5 - 2\sqrt{6}) = 10. Then, y^2 = 10 - 2 = 8. Thus, y = \sqrt{8} = 2\sqrt{2}. This completely avoids finding the square root of a surd! Common Pitfall: A frequent mistake is plugging the raw value of x directly into the fractional expression without simplifying it first. This leads to messy rationalization of a complex denominator like \sqrt{5 + 2\sqrt{6}}, which consumes a lot of exam time and often leads to arithmetic errors. Final Answer: Therefore, the correct answer is 2\sqrt{2}.

Question 8

If $x = 2 + 2^{\frac{2}{3}} + 2^{\frac{1}{3}}$, then the value of $x^3 - 6x^2 + 6x$ is

Accepted Answer

Concept & Strategy: When faced with evaluating a higher-degree polynomial given a complex surd value for x, direct substitution is highly inefficient. The strategy is to isolate the fractional power terms on one side, cube both sides to eliminate the fractions, and algebraically manipulate the result to match the target polynomial expression. (a + b)^3 = a^3 + b^3 + 3ab(a + b) Step-by-Step Solution: • Given: x = 2 + 2^{(2/3)} + 2^{(1/3)} Expression to find: x^3 - 6x^2 + 6x • Calculation / Deduction: 1. Move the integer part to the left side to isolate the fractional powers: x - 2 = 2^{(2/3)} + 2^{(1/3)} 2. Cube both sides to eliminate the fractional exponents: (x - 2)^3 = (2^{(2/3)} + 2^{(1/3)})^3 3. Expand the left side using (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3: x^3 - 3(x^2)(2) + 3(x)(2^2) - 2^3 = x^3 - 6x^2 + 12x - 8 4. Expand the right side using (a + b)^3 = a^3 + b^3 + 3ab(a + b): (2^{(2/3)})^3 + (2^{(1/3)})^3 + 3(2^{(2/3)})(2^{(1/3)})(2^{(2/3)} + 2^{(1/3)}) 2^2 + 2^1 + 3(2^{(2/3) + (1/3)})(x - 2) *(Note: We substituted (x-2) back for the sum term)* 4 + 2 + 3(2^1)(x - 2) = 6 + 6(x - 2) = 6 + 6x - 12 = 6x - 6 5. Equate the expanded left and right sides: x^3 - 6x^2 + 12x - 8 = 6x - 6 6. Rearrange the terms to match the required expression x^3 - 6x^2 + 6x: x^3 - 6x^2 + 12x - 6x = 8 - 6 x^3 - 6x^2 + 6x = 2 Exam Strategy & Shortcut: Pattern Recognition: Questions of the format x = a + a^{2/3} + a^{1/3} frequently ask for the value of x^3 - 3ax^2 + 3a^2x. Notice how the coefficients -6 and +6 in the target expression x^3 - 6x^2 + 6x don't perfectly align with the standard binomial expansion (which yields 12x). By carefully executing the "isolate and cube" method, you systematically force the target polynomial to appear on the left, leaving only a constant integer on the right. While there isn't a 5-second shortcut, recognizing this exact structural pattern prevents you from wasting time on brute-force substitution. Common Pitfall: A major pitfall is expanding the right side (2^{2/3} + 2^{1/3})^3 and leaving the 3ab(a+b) term entirely in its fractional form. Failing to recognize that the (a+b) component can be neatly replaced with (x-2) leads to an algebraic dead end where the terms never simplify into integers. Final Answer: Therefore, the correct answer is 2.

Question 9

Find the value of $(-2)^5 	imes (2)^{-5} 	imes (3)^3$

Accepted Answer

Concept & Formula: This problem tests the basic laws of indices, specifically dealing with negative bases raised to odd powers, and the product rule for identical bases with different exponents. (-a)^{odd} = -a^{odd} a^m × a^n = a^{m+n} Step-by-Step Solution: • Given: Expression to evaluate: (-2)^5 × (2)^{-5} × (3)^3 • Calculation / Deduction: 1. First, handle the negative sign in the first term. Since a negative number is raised to an odd power (5), the result will be negative. (-2)^5 = - (2^5) 2. Substitute this back into the original expression: - (2^5) × (2)^{-5} × (3)^3 3. Group the terms with the same base (base 2). Apply the product rule of exponents (a^m × a^n = a^{m+n}): - (2^{5 + (-5)}) × (3)^3 - (2^0) × (3)^3 4. Apply the zero exponent rule, which states that any non-zero number raised to the power of 0 is 1 (a^0 = 1): - (1) × (3)^3 5. Finally, calculate the cube of 3 (3 × 3 × 3 = 27) and multiply by -1: -1 × 27 = -27 Exam Strategy & Shortcut: Mental Math Simplification: You can solve this mentally in 5 seconds. Recognize that (-2)^5 is simply -32 and 2^{-5} is (1/32). Multiplying a number by its exact reciprocal yields 1, and preserving the negative sign gives -1. So the entire base-2 portion simply cancels out to -1. You are left with -1 × 3^3 = -27. Common Pitfall: A common error is mishandling the negative base. Students sometimes incorrectly assume that (-2)^5 × (2)^{-5} simplifies to (-2 × 2)^{5-5} or similar flawed logic, losing track of the crucial negative sign that persists purely because the exponent is an odd number. Final Answer: Therefore, the correct answer is -27.

Question 10

The quotient when $10^{100}$ is divided by $5^{75}$ is

Accepted Answer

Concept & Formula: To divide large exponential numbers with different bases, break down the composite bases into their prime factors. This allows you to group terms with identical prime bases and clearly apply the quotient rule for exponents. (xy)^n = x^n \cdot y^n (a^m/a^n) = a^{m-n} Step-by-Step Solution: • Given: Expression to evaluate: \frac{10^{100}}{5^{75}} • Calculation / Deduction: 1. The base of the numerator, 10, is a composite number. Break it down into its prime factors: 10 = 2 × 5. 2. Substitute this back into the numerator: \frac{(2 × 5)^{100}}{5^{75}} 3. Apply the power of a product rule to distribute the exponent: \frac{2^{100} × 5^{100}}{5^{75}} 4. Now, group the terms with base 5 and apply the quotient rule of exponents: 2^{100} × 5^{100 - 75} 2^{100} × 5^{25} 5. This is the correct mathematical quotient in its simplest prime form, but we must manipulate it to match one of the given options. Let's factor the 2^{100} to pair it specifically with the 5^{25} term. 2^{100} = 2^{75} × 2^{25} 6. Substitute this into our expression: 2^{75} × 2^{25} × 5^{25} 7. Group the terms sharing the common exponent of 25: 2^{75} × (2 × 5)^{25} 2^{75} × 10^{25} This matches option (d). Exam Strategy & Shortcut: Reverse Engineering Options: After reaching the simplified prime form 2^{100} × 5^{25}, quickly glance at the options rather than guessing how to mathematically reconstruct the composite base. Test Option (a): 2^{25} × 10^{75} = 2^{25} × (2 × 5)^{75} = 2^{100} × 5^{75} (Incorrect). Test Option (d): 2^{75} × 10^{25} = 2^{75} × (2 × 5)^{25} = 2^{100} × 5^{25} (Correct!). Testing the options directly against your simplified prime-factored result is generally faster and less error-prone than intuitively rebuilding the expression. Common Pitfall: A frequent mistake is attempting to blindly divide the exponents directly (e.g., subtracting 75 from 100 to get an exponent of 25) without aligning the bases first. This leads students to incorrectly choose option (b) 10^{25}, completely ignoring the base 2 component that was separated during division. Final Answer: Therefore, the correct answer is 2^{75} × 10^{25}.

Question 11

The exponential form of $\sqrt{\sqrt{2} 	imes \sqrt{3}}$ is

Accepted Answer

Concept & Formula: To find the exponential form of nested surds, you must apply the product rule for radicals and then convert the radical notation into fractional exponents. \sqrt{a} × \sqrt{b} = \sqrt{a × b} \sqrt[n]{x} = x^{(1/n)} Step-by-Step Solution: • Given: The expression is \sqrt{\sqrt{2} × \sqrt{3}}. • Calculation / Deduction: 1. First, simplify the innermost part of the expression by multiplying the two square roots: \sqrt{2} × \sqrt{3} = \sqrt{2 × 3} = \sqrt{6} 2. Substitute this back into the original expression: \sqrt{\sqrt{6}} 3. Convert the inner square root to a fractional exponent: \sqrt{6^{(1/2)}} 4. Convert the outer square root to a fractional exponent: (6^{(1/2)})^{(1/2)} 5. Apply the power of a power rule (x^m)^n = x^{m × n}: 6^{(1/2) × (1/2)} = 6^{(1/4)} Exam Strategy & Shortcut: Direct Transformation: You can immediately rewrite nested square roots as a 4th root. Since \sqrt{\sqrt{x}} = \sqrt[4]{x} = x^{(1/4)}, you simply multiply the inner values (2 × 3 = 6) and apply the (1/4) power. You can solve this mentally in 2 seconds. Common Pitfall: A very common mistake is adding the fractional exponents instead of multiplying them when dealing with nested roots, leading students to incorrectly choose 6^1 or 6^{(1/2)}. Always multiply powers when a power is raised to another power. Final Answer: Therefore, the correct answer is 6^{(1/4)}.

Question 12

$21^x 	imes 21^{6.5} = 21^{12.4}$

Accepted Answer

Concept & Formula: This problem tests the foundational product rule of indices. When multiplying two exponential terms with the same base, you simply add their exponents. Once the bases on both sides of an equation are identical, their exponents can be set equal to each other. a^m × a^n = a^{m+n} Step-by-Step Solution: • Given: 21^x × 21^{6.5} = 21^{12.4} • Calculation / Deduction: 1. Apply the product rule of exponents to the Left Hand Side (LHS) by combining the common base of 21: 21^{x + 6.5} = 21^{12.4} 2. Since the bases on both sides of the equation are equal (21) and non-zero, their corresponding exponents must be strictly equal: x + 6.5 = 12.4 3. Solve for the unknown variable x by subtracting 6.5 from 12.4: x = 12.4 - 6.5 x = 5.9 Exam Strategy & Shortcut: Mental Arithmetic: Recognize immediately that the base 21 is irrelevant to the actual calculation. The problem instantly simplifies in your head to x + 6.5 = 12.4. Subtraction of simple decimals (12.4 - 6.5) yields 5.9 directly. Common Pitfall: Students rushing through the exam might mistakenly divide 12.4 by 6.5, confusing the exponent rules for multiplication with the rules for power of a power. Always remember that multiplication of bases means addition of exponents. Final Answer: Therefore, the correct answer is 5.9.

Question 13

$\frac{5.4 \div 3 	imes 16 \div 2}{18 \div 5 	imes 6 \div 3}$

Accepted Answer

Concept & Strategy: This question evaluates your strict adherence to the BODMAS / PEMDAS rule (Order of Operations). For sequences containing only multiplication and division, you must evaluate the expression strictly from left to right. Evaluate \div and × strictly from Left to Right Step-by-Step Solution: • Given: Numerator: 5.4 \div 3 × 16 \div 2 Denominator: 18 \div 5 × 6 \div 3 • Calculation / Deduction: 1. Evaluate the numerator strictly left to right: • Step 1: 5.4 \div 3 = 1.8 • Step 2: 1.8 × 16 = 28.8 • Step 3: 28.8 \div 2 = 14.4 • Numerator result = 14.4 2. Evaluate the denominator strictly left to right: • Step 1: 18 \div 5 = 3.6 • Step 2: 3.6 × 6 = 21.6 • Step 3: 21.6 \div 3 = 7.2 • Denominator result = 7.2 3. Divide the simplified numerator by the simplified denominator: (14.4/7.2) = 2 Exam Strategy & Shortcut: Fractional Conversion: Instead of dealing with decimals, convert division signs into multiplication by reciprocals. This allows you to cancel terms efficiently. Numerator: 5.4 × (1/3) × 16 × (1/2) = 5.4 × (16/6) = 5.4 × (8/3) = 1.8 × 8 = 14.4 Denominator: 18 × (1/5) × 6 × (1/3) = (18 × 6/15) = (108/15) = 7.2 Result: (14.4/7.2) = 2. Common Pitfall: The most common mistake is performing multiplication before division regardless of their order. For example, in the denominator 18 \div 5 × 6 \div 3, a student might wrongly calculate 5 × 6 = 30 first, completely ruining the evaluation sequence and resulting in an incorrect fraction. Final Answer: Therefore, the correct answer is 2.

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