Aptitude question and answer on Surds and indices

In this section we are going to discuss about surds and indices problems. Not just the overview of the topic, but also you are going to learn the important formulas on surds and indices along with explanation which is easy to understand.

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