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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   \({-1} \over x\)
 

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.

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

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:
\(\{(16)^{3/2} + (16)^{-3/2}\}\)

=> \(\{(4^2)^{3/2} + (4^2)^{-3/2}\}\)

=> \(4^3+ 4^{-3}\) = \(4^3+ {1 \over 4^{3}}\) = 

=> \(64+ {1 \over 64}\) = \(4079 \over 64\)

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.
 

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



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

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