# Surds and Indices

$\text{1.&space;}&space;\left&space;(&space;100&space;+&space;10&space;+&space;1&space;\right&space;)^0&space;=&space;\,\,&space;?$

A. 0

B. 111

C. 1

D. 10

Ans: C

Explanation:

$\left&space;(&space;100&space;+&space;10&space;+&space;1&space;\right&space;)^0&space;=&space;\left&space;(&space;111&space;\right&space;)^0&space;=&space;1&space;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,&space;\left&space;[&space;a^0&space;=&space;1&space;\right&space;]$

$\text{2.&space;}&space;49&space;\times&space;49&space;\times&space;49&space;\times&space;49&space;=&space;7^?$

A. 4

B. 6

C. 7

D. 8

Ans:  D

Explanation:

$49&space;\times&space;49&space;\times&space;49&space;\times&space;49&space;=&space;7^?$

$\left&space;(&space;7^2&space;\times&space;7^2&space;\times&space;7^2&space;\times&space;7^2&space;\right&space;)&space;=&space;7^?$

$7^{\left&space;(&space;2&space;+&space;2&space;+&space;2&space;+&space;2&space;\right&space;)}&space;=&space;7^?$

$7^8&space;=&space;7^?$

$\therefore&space;\,\,&space;?&space;=&space;8$

$\text{3.&space;}&space;\left&space;(&space;1000&space;\right&space;)^7&space;\div&space;10^{18}&space;=&space;\,\,&space;?$

A. 10

B. 100

C. 1000

D. 10000

Ans: C

Explanation:

$\left&space;(&space;1000&space;\right&space;)^7&space;\div&space;10^{18}&space;=&space;\frac{\left&space;(&space;1000&space;\right&space;)^7}{10^{18}}$

$=&space;\frac{\left&space;(&space;10^3&space;\right&space;)^7}{10^{18}}$

$=&space;\frac{10^{21}}{10^{18}}$

$=&space;10^{21}&space;\times&space;10^{-18}$

$=&space;10^3$

$=&space;1000$

$\text{4.&space;}&space;\left&space;(&space;256&space;\right&space;)^{0.16}&space;\times&space;\left&space;(&space;256&space;\right&space;)^{0.09}&space;=&space;\,\,&space;?$

A. 4

B. 16

C. 125

D. 625

Ans: A

Explanation:

$\left&space;(&space;256&space;\right&space;)^{0.16}&space;\times&space;\left&space;(&space;256&space;\right&space;)^{0.09}&space;=&space;\left&space;(&space;256&space;\right&space;)^{\left&space;(&space;0.16&space;\,&space;+&space;\,&space;0.09&space;\right&space;)}$

$=&space;\left&space;(&space;256&space;\right&space;)^{0.25}$

$=&space;\left&space;(&space;256&space;\right&space;)^\frac{25}{100}$

$=&space;\left&space;(&space;256&space;\right&space;)^\frac{1}{4}$

$=&space;\left&space;(&space;4^4&space;\right&space;)^\frac{1}{4}$

$=&space;4$

$\text{5.&space;The&space;value&space;of&space;}&space;\left&space;(&space;256&space;\right&space;)^\frac{5}{4}&space;\text&space;\,\text{is}$

A. 256

B. 512

C. 756

D. 1024

Ans: D

Explanation:

$\left&space;(&space;256&space;\right&space;)^\frac{5}{4}&space;=&space;\left&space;(&space;4^4&space;\right&space;)^\frac{5}{4}&space;=&space;4^{4&space;\times&space;\frac{5}{4}}&space;=&space;4^5&space;=&space;1024$

$\text{6.&space;If&space;}\sqrt{2^n}&space;=&space;64,&space;\text{then&space;the&space;value&space;of&space;n&space;is}$

A. 6

B. 8

C. 10

D. 12

Ans: D

Explanation:

$\sqrt{2^n}&space;=&space;64$

$\sqrt{2^n}&space;=&space;2^6$

$\left&space;(&space;2^n&space;\right&space;)^\frac{1}{2}&space;=&space;2^6&space;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,&space;\left&space;[&space;\sqrt{a}&space;=&space;a^\frac{1}{2}\right&space;]$

$2^\frac{n}{2}&space;=&space;2^6$

$\text{Since&space;bases&space;are&space;equal&space;exponents&space;are&space;equal.}$

$\frac{n}{2}&space;=&space;6$

$n&space;=&space;12$

$\text{7.&space;If&space;}&space;\,&space;5^{\left&space;(&space;x&space;\,&space;+&space;\,&space;3&space;\right&space;)}&space;=&space;\left&space;(&space;25&space;\right&space;)^{\left&space;(&space;3x&space;\,&space;-&space;\,&space;4&space;\right&space;)}&space;\text{then&space;the&space;value&space;of}&space;\,&space;\,&space;x&space;\,\,&space;\text{is}$

$\text{A.&space;}&space;\frac{5}{11}$

$\text{B.&space;}&space;\frac{11}{3}$

$\text{C.&space;}&space;\frac{5}{3}$

$\text{D.&space;}&space;\frac{11}{5}$

Ans: D

Explanation:

$5^{\left&space;(&space;x&space;\,&space;+&space;\,&space;3&space;\right&space;)}&space;=&space;\left&space;(&space;25&space;\right&space;)^{\left&space;(&space;3x&space;\,&space;-&space;\,&space;4&space;\right&space;)}$

$5^{\left&space;(&space;x&space;\,&space;+&space;\,&space;3&space;\right&space;)}&space;=&space;\left&space;(&space;5^2&space;\right&space;)^{\left&space;(&space;3x&space;\,&space;-&space;\,&space;4&space;\right&space;)}$

$5^{\left&space;(&space;x&space;\,&space;+&space;\,&space;3&space;\right&space;)}&space;=&space;5^{\left&space;(&space;6x&space;\,&space;-&space;\,&space;8&space;\right&space;)}$

$\text{Since&space;bases&space;are&space;equal&space;exponents&space;are&space;equal.}$

$x&space;+&space;3&space;=&space;6x&space;-&space;8$

$5x&space;=&space;11$

$x&space;=&space;\frac{11}{5}$

$\text{8.&space;If&space;}&space;5^a&space;=&space;3125&space;\,&space;\,&space;\text{then&space;the&space;value&space;of&space;}&space;5^{\left&space;(&space;a&space;\,&space;-&space;\,&space;3&space;\right&space;)}&space;\,&space;\text{is&space;}$

A. 25

B. 625

C. 125

D. 5

Ans: A

Explanation:

$5^a&space;=&space;3125$

$5^a&space;=&space;5^5$

$a&space;=&space;5$

$\therefore&space;5^{\left&space;(&space;a&space;\,&space;-&space;\,&space;3&space;\right&space;)}&space;=&space;5^{\left&space;(&space;5&space;\,&space;-&space;\,&space;3&space;\right&space;)}&space;=&space;5^2&space;=&space;25$

$\text{9.&space;If&space;}&space;\frac{9^n&space;\times&space;3^5&space;\times&space;\left&space;(&space;27&space;\right&space;)^3}{3&space;\times&space;\left&space;(&space;81&space;\right&space;)^4}&space;=&space;27&space;\,\,&space;\text{then&space;the&space;value&space;of&space;n&space;is}$

A. 1

B. 2

C. 3

D. 4

Ans: C

Explanation:

$\frac{9^n&space;\times&space;3^5&space;\times&space;\left&space;(&space;27&space;\right&space;)^3}{3&space;\times&space;\left&space;(&space;81&space;\right&space;)^4}&space;=&space;27$

$\frac{\left&space;(&space;3^2&space;\right&space;)^n&space;\times&space;3^5&space;\times&space;\left&space;(&space;3^3&space;\right&space;)^3}{3&space;\times&space;\left&space;(&space;3^4&space;\right&space;)^4}&space;=&space;3^3$

$\frac{3^{2n}&space;\times&space;3^5&space;\times&space;3^9}{3&space;\times&space;3^{16}}&space;=&space;3^3$

$\frac{3^{2n}&space;\times&space;3^{14}}{3^{17}}&space;=&space;3^3$

$3^{2n}&space;\times&space;3^{14}&space;\times&space;3&space;^{-17}&space;=&space;3^3$

$3&space;\,&space;^{\left&space;(&space;2n&space;\,&space;+&space;\,&space;14&space;\,&space;-&space;\,&space;17&space;\right&space;)}&space;=&space;3^3$

$3&space;\,&space;^{\left&space;(&space;2n&space;\,&space;-&space;\,&space;3&space;\right&space;)}&space;=3^3$

$\text{Since&space;bases&space;are&space;equal&space;exponents&space;are&space;equal.}$

$2n&space;-&space;3&space;=&space;3$

$2n&space;=&space;6$

$n&space;=&space;3$

$\text{10.&space;The&space;value&space;of&space;}&space;\frac{1}{\left&space;(&space;216&space;\right&space;)^{-\frac{2}{3}}}&space;+&space;\frac{1}{\left&space;(&space;256&space;\right&space;)^{-\frac{3}{4}}}&space;+&space;\frac{1}{\left&space;(&space;32&space;\right&space;)^{-\frac{1}{5}}}&space;\,\,&space;\text{is}$

A. 92

B. 96

C. 98

D. 102

Ans: D

Explanation:

$\frac{1}{\left&space;(&space;216&space;\right&space;)^-\frac{2}{3}}&space;+&space;\frac{1}{\left&space;(&space;256&space;\right&space;)^-\frac{3}{4}}&space;+&space;\frac{1}{\left&space;(&space;32&space;\right&space;)^-\frac{1}{5}}&space;=&space;\frac{1}{\left&space;(&space;6^3&space;\right&space;)^-\frac{2}{3}}&space;+&space;\frac{1}{\left&space;(&space;4^4&space;\right&space;)^-\frac{3}{4}}&space;+&space;\frac{1}{\left&space;(&space;2^5&space;\right&space;)^-\frac{1}{5}}$

$=&space;\frac{1}{6\,&space;^{3&space;\,&space;\times&space;\,&space;\left&space;(&space;-\frac{2}{3}&space;\right&space;)}}&space;+&space;\frac{1}{4&space;\,&space;^{4&space;\,&space;\times&space;\,&space;\left&space;(&space;-\frac{3}{4}&space;\right&space;)}}&space;+&space;\frac{1}{2&space;\,&space;^{5&space;\,&space;\times&space;\,&space;\left&space;(&space;-\frac{1}{5}&space;\right&space;)}}$

$=&space;\frac{1}{6^{-2}}&space;+&space;\frac{1}{4^{-3}}&space;+&space;\frac{1}{2^{-1}}$

$=&space;6^2&space;+&space;4^3&space;+&space;2^1$

$=&space;36&space;+&space;64&space;+&space;2$

$=&space;102$