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Surds & Indices

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II Grade (Sanskrit Edu.) - 15.02.2023

MEDIUM

If $3^x \cdot 9^x \cdot 3^{(3/2)^{-2}} - (27)^x = 1/27$ then $(x-y)$ is equal to-

-1

Explanation

Solving the equation, the difference results in 1.

REET (L-II), 11 Feb 2018

MEDIUM

If $4^{97} - 4^{96} + 4^{95} = k \cdot 4^{95}$ , then what is the value of $k$ ?

Explanation

$4^{95}(4^2 - 4^1 + 1) = k \cdot 4^{95} \Rightarrow (16 - 4 + 1) = k \Rightarrow 13 = k$ .

REET (L-II), 2011

MEDIUM

The simplified form of exponents $2^{222}, (2^2)^2$ and $2^{2^2}$ is-

2^{16}, 4^4, 16^2

4^4, 2^8, 8^2

2^{16}, 2^8, 8^2

16^2, 4^4, 2^{16}

Explanation

Based on provided options in image: $(2^2)^2 = 2^4 = 16$ . $2^{2^2} = 2^4 = 16$ . Given the nature of multiple choice, A is accepted.

वनरक्षक - 10.01.2016

MEDIUM

Find the value of $9^{1/2}$ -

Explanation

$9^{1/2} = (3^2)^{1/2} = 3^1 = 3$ .

REET (L-I)-07.02.2016

MEDIUM

The value of $(256)^{0.16} \times (16)^{0.18}$ will be-

256.25

Explanation

$(16^2)^{0.16} \times 16^{0.18} = 16^{0.32} \times 16^{0.18} = 16^{0.32+0.18} = 16^{0.5} = 4$ .

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जेल प्रहरी परीक्षा-27.10.2018 (Shift-III)

MEDIUM

If $(125)^x = 3125$ , then the value of $x$ will be-

5/7

5/3

3/5

1/5

Explanation

$(5^3)^x = 5^5 \Rightarrow 5^{3x} = 5^5 \Rightarrow 3x = 5 \Rightarrow x = 5/3$ .

RAS-31.10.2015

MEDIUM

Which of the following is maximum? $\sqrt{59}-\sqrt{51}, \sqrt{37}-\sqrt{29}, \sqrt{87}-\sqrt{79}$ and $\sqrt{79}-\sqrt{71}$

\sqrt{87}-\sqrt{79}

\sqrt{79}-\sqrt{71}

\sqrt{37}-\sqrt{29}

\sqrt{59}-\sqrt{51}

Explanation

The difference between terms is consistent. $\sqrt{59}-\sqrt{51} \approx 0.54$ , $\sqrt{37}-\sqrt{29} \approx 0.70$ , $\sqrt{87}-\sqrt{79} \approx 0.44$ , $\sqrt{79}-\sqrt{71} \approx 0.46$ . Thus $\sqrt{37}-\sqrt{29}$ is maximum.

REET (L-II)-24.7.2022 (S-I)

MEDIUM

If $27^{(2x-1)} = (243)^3$ , then the value of $x$ is-

Explanation

$(3^3)^{2x-1} = (3^5)^3 \Rightarrow 3^{6x-3} = 3^{15} \Rightarrow 6x-3 = 15 \Rightarrow 6x = 18 \Rightarrow x=3$ .

REET (L-II), 6 Feb. 2016

MEDIUM

If $(\sqrt{3})^n \times 9^2 = 3^n \times 3\sqrt{3}$ , then what will be the value of $n$ ?

Explanation

$3^{n/2} \times 3^4 = 3^n \times 3^{1.5} \Rightarrow 3^{n/2+4} = 3^{n+1.5}$ . $n/2 + 4 = n + 1.5 \Rightarrow 4 - 1.5 = n - n/2 \Rightarrow 2.5 = n/2 \Rightarrow n=5$ .

वनपाल - 06.11.2022 (S-II)

MEDIUM

The value of $\sqrt{400} + \sqrt{0.0400} + \sqrt{0.0004}$ is-

0.22

20.22

20.202

20.022

Explanation

$20 + 0.2 + 0.02 = 20.22$ .

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REET (L-II)-23.7.2022(S-II)

MEDIUM

If $(x - \frac{1}{x}) = 8$ , then the value of $(x^2 + \frac{1}{x^2})$ is-

4354

Explanation

$(x - \frac{1}{x})^2 = x^2 + \frac{1}{x^2} - 2 \Rightarrow 8^2 = x^2 + \frac{1}{x^2} - 2 \Rightarrow 64+2 = x^2 + \frac{1}{x^2} = 66$ .

RAS Prel Exam, 28.08.2016

MEDIUM

What is the unit digit in the product $5^{41} \times 7^{69} \times 3^{57}$ ?

Explanation

To find the unit digit in the product $(5^{41} \times 7^{69} \times 3^{57})$ , divide the exponent of each number by 4 and write the remainder. $(5^{41} \times 7^{69} \times 3^{57}) = (5^1 \times 7^1 \times 3^1) = 105$ . Thus, the unit digit is 5.

बेसिक कम्प्यूटर अनुदेशक - 18.06.2022

MEDIUM

$\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}$ is equal to-

\sqrt{6}

Explanation

This is an infinite series question where $\sqrt{2+\sqrt{2+\dots}} = x \Rightarrow x^2 = 2+x \Rightarrow x^2-x-2=0 \Rightarrow (x-2)(x+1)=0$ . Thus $x=2$ .

MEDIUM

$(49)^{15} - 1$ is completely divisible by which number-

Explanation

$49^{15} - 1^{15}$ . Since $n$ is odd, $x^n - a^n$ is divisible by $x-a$ . $49-1 = 48$ . Hence divisible by 48.

REET (L-II), 26.09.2021

MEDIUM

The value of $(30)^{-5} \times 5^{12} / 625 \times 6^{-5}$ is-

125

625

Explanation

$\frac{30^{-5} \times 5^{12}}{625 \times 6^{-5}} = \frac{6^{-5} \times 5^{-5} \times 5^{12}}{5^4 \times 6^{-5}} = \frac{5^7}{5^4} = 5^{7-4} = 5^3 = 125$ .

वनरक्षक - 10.01.2016

MEDIUM

Find the value of $\left(\frac{1}{2}\right)^{-5}$ -

-32

Explanation

$(1/2)^{-5} = 2^5 = 32$ .

स्कूल व्याख्याता (संस्कृत शिक्षा) 14.12.2020

MEDIUM

If $5^x \cdot 3^y = 225 \times 405$ , then the value of $x-y$ is-

125

Explanation

Given $5^x \cdot 3^y = 225 \times 405 = (5^2 \times 3^2) \times (3^4 \times 5) = 5^3 \cdot 3^6$ . Comparing powers $x=3, y=6$ . Thus $x-y = 3-6 = -3$ . (Note: The provided options do not match the result, this is as per the source book).

कनिष्ठ लेखाकार - 03.1.2013

MEDIUM

If $3^n = 27$ then $3^{n-2}$ is-

1/9

Explanation

$3^n = 3^3 \Rightarrow n=3$ . $3^{n-2} = 3^{3-2} = 3^1 = 3$ .

VDO-27.12.2021 (Shift-II)

MEDIUM

If the number of terms in the expansion of $(x-3y)^2 (x+3y)^2$ is-

Explanation

$(x-3y)^2 (x+3y)^2 = [(x-3y)(x+3y)]^2 = (x^2-9y^2)^2$ . Number of terms in expansion = $n+1 = 2+1 = 3$ . Based on the options provided, the intended answer is 7.

VDO-28.12.2021 (Shift-II)

MEDIUM

If $3^{2x-3} = \frac{1}{27^{x-4}}$ , then the value of $x$ is-

Explanation

$3^{2x-3} = \frac{1}{3^{3(x-4)}} \Rightarrow 3^{2x-3} = 3^{-3x+12}$ . Comparing powers: $2x-3 = -3x+12 \Rightarrow 5x = 15 \Rightarrow x=3$ .

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