How To Solve Surds And Indices Questions Quickly

March 6, 2024

Solve Surds and Indices Questions Quickly

On this page you will get to know about, how to solve surds and indices quickly and easy using the formulas and rules. 

How to Solve Surds and Indices Questions Quickly

How to Solve Surds and Indices Ques. Quickly :

  • Surds: Number which cannot be expressed in the fraction form of two integers is called as surd. For example: \sqrt{2}
  • Indices: Indices refers to the power to which a number is raised. For example; 2²

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Sample Solutions Regarding the Rules of Surds and Indices :

Indices Multiplication rules:-

  • Multiplication rule with same base
    a n ⋅ a m = a(m+n)
    Example:

    23 ⋅ 24 = 2(3+4) = 27 = 2⋅2⋅2⋅2⋅2⋅2⋅2 = 128

Surds Multiplication rules:-

  • Multiplication rule with same indices
    an ⋅ bn = (a ⋅ b)n
    Example:
    32 ⋅ 22 = (3⋅2)2 = 36

Indices Division rules:-

  • Division rule with same indices
    an / bn = (a / b)n
    Example: 93 / 33 = (9/3)3 = 27

Surds Division rules:-

  • Division rule with same base
    am / an = a(m-n)
    Example: 35 / 33 = 3(5-3) = 9

Surds and Indices Power rules

Power rule 1

  • (an)m = a(n.m)
    Example:
    (23)2 = 2(3.2) = 26 = 2⋅2⋅2⋅2⋅2⋅2 = 64

Power rule 2

  • anm= a(nm) =
    Example:
    232 = 2(32) = 29 = 2⋅2⋅2⋅2⋅2⋅2⋅2⋅2⋅2 = 512

Power rule 3

  • n√a =a(1/n)
    Example:
    3√27 = 27(1/3) = 3

Type 1: How to Solve Surds and Indices Quickly- Simplify the expression

Question 1. Find the value of 7-25 – 7-26

Options:

A. 6 × 7-26

B. 6 × 7-25

C. 7 × 7-25

D. 7 × 7-26

Solution:    7-25 – 7-26 = \frac{1}{7^{25}} – \frac{1}{7^{26}}

\frac{7 – 1}{7^{26}} = 6 × 7-26

Correct option: A

Question 2. Simplify (256)^ \frac{3}{4}

Options:

A. 16

B. 12

C. 256

D. 64

Solution:    (256)^ \frac{3}{4} = (44)^ \frac{3}{4}= 43 = 64

Correct option: D

Question 3. Find the value of 8112 ÷ 8110

Options:

A. 72

B. 64

C. 81

D. 49

Solution:    We know,

\frac{a^m}{a^n} = am-n

= 8 (112 – 110) = 82 = 64

Correct option: B

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

Question 1. If 4x + 4x + 1 = 80, then the value of xx  is

Options:

A. 16

B. 9

C. 25

D. 4

Solution:     4x(1 + 4) = 80

4x  * 5 = 80

4x = \frac{80}{5}

4x = 16

 x = 2

 xx = 22 = 4

Correct option: D

Question 2. If 2a = 3 \sqrt{32}, then a is equal to:

Options:

A. \frac{1}{3}

B. 4

C. \frac{5}{3}

D. \frac{1}{2}

Solution:    Given value 2a

3 \sqrt{32}

2a = (32)^ \frac{1}{3}

2a = (25)^ \frac{1}{3}

2a = (2)^ \frac{5}{3}

a = \frac{5}{3}

Correct option: C

Question 3. If x and y are whole numbers such that xy = 169, then find the value of (x – 1)y + 1

Options:

A. 1331

B. 2744

C. 1728

D. 729

Solution:    169 = 132

So, the value of x = 13 and y = 2

Now, (x – 1)y + 1

= (13 – 1)2+1

= (12)3 

=1728

Correct option: C

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