Surds & Indices — CAT Previous-Year Questions

If (a + b√n) is the positive square root of (29 - 12√5), where a and b are integers, and n is a natural number, then the maximum possible value of (a + b + n) is

The sum of all real values of k for which ${\left(\frac{1}{8}\right)}^k$ × ${\left(\frac{1}{32768}\right)}^{\frac{1}{2}}$ = $\left(\frac{1}{8}\right)$ × ${\left(\frac{1}{32768}\right)}^{\frac{1}{k}}$, is

If m and n are natural numbers such that n > 1, and m = 2²⁵ × 3⁴⁰, then m - n equals

If ${\left(x+6\sqrt{2}\right)}^{\frac{1}{2}}$ - ${\left(x-6\sqrt{2}\right)}^{\frac{1}{2}}$ = $2\sqrt{2}$, then x equals

(a + b√3)² = 52 + 30√3, where a and b are natural numbers, then a + b equals

If 3^a = 4, 4^b = 5, 5^c = 6, 6^d = 7, 7^e = 8 and 8^f = 9, then the value of the product abcdef is

If $\sqrt{5x+9}$ + $\sqrt{5x-9}$ = 3(2 + √2), then find the value of $\sqrt{10x+9}$?

The sum of all possible values of x satisfying the equation ${2^{4x}}^2$ - $2^{2x^2+x+16}$ + $2^{2x+30}$ = 0, is

Let a, b, m and n be natural numbers such that a > 1 and b > 1. If a^mbⁿ = 144¹⁴⁵, then the largest possible value of n - m is

Let n be any natural number such that 5^n-1 < 3ⁿ⁺¹. Then, the least integer value of m that satisfies 3ⁿ⁺¹ < 2^n+m for each such n, is?

If ${\left(\sqrt{\frac{7}{5}}\right)}^{3x-y}$ = $\frac{875}{2401}$ and ${\left(\frac{4a}{b}\right)}^{6x-y}$ = ${\left(\frac{2a}{b}\right)}^{y-6x},$ for all non-zero real values of a and b, then the value of x + y is

For all possible integers n satisfying 2.25 ≤ 2 + 2ⁿ⁺² ≤ 202, the number of integer values of 3 + 3ⁿ⁺¹ is

In ‘n’ is a positive integer such that $\left(\sqrt[7]{10}\right)$${\left(\sqrt[7]{10}\right)}^2$...${\left(\sqrt[7]{10}\right)}^n$ > 999, then the smallest value of n is

The number of real-valued solutions of the equation 2^x + 2^-x = 2 – (x – 2)² is

If x = (4096)^7+4√3, then which of the following equals 64?

If a, b, c are non-zero and 14^a = 36^b = 84^c, then $6\mathrm{b}\left(\frac{1}{\mathrm{c}}-\frac{1}{\mathrm{a}}\right)$ is equal to

How many pairs (a, b) of positive integers are there such that a ≤ b and ab = 4²⁰¹⁷?

If m and n are integers such that (√2)¹⁹ 3⁴ 4² 9^m 8ⁿ = 3ⁿ 16^m (64)^1/4, then m is

If 5.55^x = 0.555^y = 1000, then the value of (1/x) − (1/y) is

If 5^x - 3^y = 13438 and 5^x-1 + 3^y+1 = 9686, then x + y equals

Given that x²⁰¹⁸y²⁰¹⁷ = 1/2 and x²⁰¹⁶y²⁰¹⁹ = 8, the value of x² + y³ is

If N and x are positive integers such that N^N = 2¹⁶⁰ and N² + 2^N is an integral multiple of 2^x, then the largest possible x is

If a and b are integers of opposite signs such that (a + 3)² : b² = 9 : 1 and (a - 1)²: (b - 1)² = 4 : 1, then the ratio a² : b² is:

If 9^2x–1 – 81^x-1 = 1944, then x is

If $9^{x-\frac{1}{2}}$ - 2^{2x - 2} = 4^x - 3^{2x - 3}, then x is

If x = −0.5, then which of the following has the smallest value?

Which among 2^1/2, 3^1/3, 4^1/4 , 6^1/6  and 12^1/12  is the largest ?

What are the values of x and y that satisfy both the equations?

2^0.7x . 3^{−1. 25y} = $\frac{8\sqrt{6}}{27}$

4^0.3x . 9^0.2y = 8 . (81)^1/5

Let x = $\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-....to infinity.}}}}$ Then x equals

Let $y=\frac{1}{2+{\displaystyle \frac{1}{3+{\displaystyle \frac{1}{2+{\displaystyle \frac{1}{3+....}}}}}}}$

What is the value of y?

If pqr = 1 then $\frac{1}{1+p+q^{-1}}+\frac{1}{1+q+r^{-1}}+\frac{1}{1+r+p^{-1}}$ is equivalent to

Which of the following is true?

A, B and C are defined as follows.

A = 2.000004 ÷ [(2.000004)² + (4.000008)]

B = 3.000003 ÷ [(3.000003)² + (9.000009)]

C = 4.000002 ÷ [(4.000002)² + (8.000004)]

Which of the following is true about the values of the above three expressions?

2⁷³ – 2⁷² – 2⁷¹ is the same as