Surds And Indices Tips, Tricks And Shortcuts

Question of Surds and Indices with Explanation

Type 1: Surds and Indices -Simplify the expression

Question . Find the value (1728)^{( –  \frac{2}{3})}

Options.

A. \frac{1}{144}

B. 144

C. – \frac{1}{144}

D. \frac{1}{12}

Solution:     Cube root of 1728 is 12.

Therefore,(12)^{–3( \frac{1}{3})}

(12) ^{– 3 × ( \frac{2}{3})} = 12^–2

We know that, a^-n = ^{\frac{1}{a^n}}

Therefore, 12^–2 = (\frac{1}{12})^2 = \frac{1}{144}

Correct option: A

Type 2 : Surds and Indices- Find the value of x

Question . Find the value of x  If ( \frac{p}{q})^x-1 = ( \frac{q}{p})^x-3

Options.

A. 3

B. 2

C. 1

D. -2

Solution:    ( \frac{p}{q})^x-1 = ( \frac{q}{p})^x-3

( \frac{p}{q})^x-1 = ( \frac{p}{q})^-(x-3)

( \frac{p}{q})^x-1 = ( \frac{p}{q})^3-x

x-1 = 3-x

x = 2

Correct option: B

Type 3 : When base is same 

Question : If 5^{a} = 3125, then the value of 5^{(a – 3)} is:

A. 25

B. 125

C. 625

D. 1625

Solution: 

5^{a} = 3125

5^{a} = 5^{5}

a = 5.

5^{(a – 3)} = 5^{(5 – 3)}

5^{2} = 25.

Question : A person wants to know answer of question asked in annual exams as follows: 3^{0} + \frac{3^{-1}}{3^{-1}}- 3^{0} is simplified to

(1) –2
(2) –1
(3) 1
(4) 2

Solution :

Expression = 3^{0} + \frac{3^{-1}}{3^{-1}} – 3^{0}

\frac{1 +\frac{1}{3}}{\frac{1}{3} -1}

\frac{\frac{(3+1)}{3} }{\frac{(1-3)}{3}}

\frac{4}{3} \times -\frac{3}{2}
= –2

Question : Simplify the equation (256)^{0.16} × (4)^{0.36} is equal to

(1) 64
(2) 16
(3) 256.25
(4) 4

Solution :

(256)^{0.16} \times (4)^{0.36}

(44)^{0.16} \times 4^{0.36}

4^{0.64} \times 4^{0.36}

(4)^{0.64 + 0.36} = 4