Question 1

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 2

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 3

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 4

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 5

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 6

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

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 7

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

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.

Surds and Indices Questions

Questions

Practice smarter