Surds And Indices Tips, Tricks And Shortcuts

March 6, 2024

Surds and Indices Tips Tricks and Shortcuts

This page contains Tips, tricks and shortcuts of Surds and Indices along with its simplification and solved question and answers.

Tips and Tricks and Shortcuts For Surds and Indices

Tips Tricks & Shortcuts for Surds and Indices

  • Surds: Numbers which can be expressed in the form \sqrt{p} + \sqrt{q} , where p and q are natural numbers and not perfect squares.
  • Indices: Indices refers to the power to which a number is raised. For example; 32
  • Here, are quick and easy tips and tricks for you to solve Surds and Indices Formulas with Power questions quickly, easily, and efficiently in competitive exams and other recruitment exams

Surds and Indices Formulas

  • (a + b)(a – b) = (a2 – b2)
  • (a + b)² = (a² + b² + 2ab)
  • (a – b)² = (a² + b²- 2ab)
  • (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
  • (a³ + b³) = (a + b)(a² – ab + b²)
  • (a³ – b³) = (a – b)(a²+ ab + b²)
  • (a³ + b³ + c³ – 3abc) = (a + b + c)(a² + b² + c² – ab – bc – ac)
  • When a + b + c = 0, then a³ + b³ + c³ = 3abc.

Surds and Indices Rules

 

Rule NameSurds RuleIndices Rule
Multiplication Rulean * bn = (a*b)nan * am = a(m+n)
Division Rulean/ bn = (a/b)nam / an = a(m-n)
Power Rule(an)m = (a)nm
n√a = a(1/n)		a(nm)) = anm
a-n = 1/(an)

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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

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