Introduction to Surds and Indices

The natural number which cannot be expressed in the form of fraction known as Surds.

For example:$ \sqrt{2}=2^{\frac{1}{2}}$ and the Indices refers to the power to which a number is raised.

Surds: Number which cannot be expressed in the fraction form of two integers is called as surd.

Indices: Indices refers to the power to which a number is raised. For example; 2²

• Multiplication rule with same base

  $a^n ⋅ a^m = a^{(m+n)}$

  Example:

  $2_{3} ⋅ 2_{4} = 2^(3+4) = 2^7 = 2⋅2⋅2⋅2⋅2⋅2⋅2 = 128$

• Multiplication rule with same indices

  $a^n ⋅ b^n = (a ⋅ b)^n$

  Example:

  $3^2 ⋅ 2^2 = (3⋅2)^2 = 36$

• Division rule with same indices

  $ \frac{a^n}{b^n} = (\frac{a}{b})^n$

  Example:

  $ \frac{9^3}{3^3} = (\frac{9}{3})^3 = 27$

• Power rule 1

  $(a^n)^m = a^{(n.m)}$

  Example:

  $(2^3)^2 = 2^{(3.2)} = 26 = 2⋅2⋅2⋅2⋅2⋅2 = 64$

• Power rule 2

  $ a^{n^m} = a^{(n^m)}$

  Example:

  $2^{3^2} = 2^{(3^2)} = 29 = 2⋅2⋅2⋅2⋅2⋅2⋅2⋅2⋅2 = 512$

1. 1. Find the value of $7^{-25} – 7^{-26}$

Options:

Solution:

Correct option: A

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

Options:

Solution:

Correct option: D

3.Find the value of $8^{112} ÷ 8^{110}$

Options:

Solution:

Correct Option: B

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

Options:

Solution:

Correct option: D

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

Options:

Solution:

Correct option: C

3.Let x and y be two positive real numbers such that $xy=1$. The minimum value of x+y is

Options:

Solution:

Correct Option: C