## Aptitude question and answer on Surds and indices

 1 What is the quotient when $(x^{-1} - 1)$ is divided by $(x-1)$ ? A.    $x$ B.    ${-1} \over x$ C.    ${-2} \over x$ D.    ${-3} \over x$ Answer : B.   ${-1} \over x$ Explanation: $(x^{-1} - 1) \over {x-1}$=$({1 \over x} - 1) \over {x-1}$=$({1-x \over x} ) \over {x-1}$=${1-x \over x} \times {1 \over {x-1}}$=${-1} \over x$ Hence, the required quotient is   ${-1} \over x$   Discuss 2 If $2^{x - 1}$ + $2^{x + 1}$ = 1280, then find the value of  x. A.    6 B.    7 C.    8 D.    9 Answer : D.  9 Explanation: $2^{x - 1}$ + $2^{x + 1}$ = 1280 => $2^{x - 1}$$(1+2^2)$=1280 =>$2^{x - 1}$=${1280 \over 5}$=256=$2^8$ => $x-1=8$ =>$x=1$ Hence, x = 9. Discuss 3 Find the value of   $[ 5 ( 8^{1\over3} + 27^{1\over3})^3]^{1\over 4}$ A.    3 B.    4 C.    5 D.    6 Answer : C.  5 Explanation: $[ 5 ( 8^{1\over3} + 27^{1\over3})^3]^{1\over 4}$ => $[ 5 \{ (2^3)^{1/3} + (3^3)^{1/3}\}^3]^{1/ 4}$ => $[ 5 (2+ 3)^3]^{1/ 4}$ =>$( 5 \times 5^3)^{1/ 4}$ =>$( 5^4)^{1/ 4}$ => 5 Discuss 4 Find the Value of $\{(16)^{3/2} + (16)^{-3/2}\}$ A.    4077/64 B.    4087/64 C.    4067/64 D.    4097/64 Answer : D.  4097/64 Explanation: $\{(16)^{3/2} + (16)^{-3/2}\}$ => $\{(4^2)^{3/2} + (4^2)^{-3/2}\}$ => $4^3+ 4^{-3}$ = $4^3+ {1 \over 4^{3}}$ =  => $64+ {1 \over 64}$ = $4079 \over 64$ Discuss 5 If $(1/5)^{3y}$ = 0.008, then find the value of $(0.25)^y$. A.    0.25 B.    0.50 C.    0.75 D.    0.1 Answer : A.  0.25 Explanation: $(1/5)^{3y}$ = 0.008 =  8/1000 =  1/125 = $({1\over5})^3$ =3y = 3 =Y = 1. $\dot{..}$ $(0.25)^y$ = $(0.25)^1$ = 0.25.   Discuss 6 Find the value of   ${(243)^{n \over 5} \times 3^{2n+1}} \over {9^n \times 3^{n-1}}$ A.    7 B.    8 C.    9 D.    10 Answer : C.  9 Explanation: ${(243)^{n \over 5} \times 3^{2n+1}} \over {(3^2)^n \times 3^{n-1}}$= ${(3^5)^{n \over 5} \times 3^{2n+1}} \over {(3^2)^n \times 3^{n-1}}$ =${3^n \times 3^{2n+1} } \over 3^{2n} \times 3^{n-1}$=${3^{n+2n+1} } \over 3^{2n+n-1}$=${3^{3n+1} } \over 3^{3n-1}$=  $3^{(3n+1)-(3n-1)}$ =$3^2$ = 9 Discuss 7 Find the value of   $6^{1 \over 3} \times \sqrt[3]{6^7} \over \sqrt[3]{6^6}$ A.    5 B.    6 C.    7 D.    8 Answer : B.   6 Explanation: $6^{2 \over 3} \times \sqrt[3]{6^7} \over \sqrt[3]{6^6}$=$6^{2 \over 3} \times(6^7)^{1 \over 3} \over (6^6)^{1 \over 3}$ =$6^{2 \over 3} \times(6^7)^{1 \over 3} \over (6^6)^{1 \over 3}$=$6^{2 \over 3} \times(6)^{7 \over 3} \over 6^2$ =$6^{2 \over 3} \times6^{({7 \over 3} -2)}$=$6^{2 \over 3} \times 6^{1 \over 3}$ =$6^1$=6 Discuss 8 Find the value of X.  $(15)^{3.5}\times (15)^x= 158$ A.    2.29 B.    2.75 C.    4.25 D.    4.5 Answer : D.  4.5 Explanation: Let $(15)^{3.5}\times (15)^x= 158$ Then, $(15)^{3.5 + x} = (15)^8$ 3.5 + x = 8 x = (8 – 3.5) x = 4.5 Discuss