### Definition

Let’s understand the definition with the help of an example:

**5³** can be expanded as **5×5×5.**

i.e. we have multiplied 5 by itself 3 times. So, index (also known as exponent or power) tells us how many times we must multiply the number by itself.

*Key Points:*

- 5 is called the base.
- 3 is called the index (also known as power or exponent) because it indicates the power to which the base ‘5’ is raised.
- 5³ is the index form (or power form) of 125.
- The plural of index is known as indices.

**Law of Indices**

**Rule 1**:** a ^{m} × a^{n} = a^{m + n}**

**Rule 2**: **a ^{m} ÷ a^{n} = a^{m ― n} **if

**(a ≠ 0)**

**Rule 3:** **a ^{0} = 1 **if

**(a ≠ 0)**

**Example**: The value of (d^{x+y} ÷ d^{x}) ÷ d^{y }= ?

*Solution:*

(d^{x+y} ÷ d^{x}) ÷ d^{y }= (d^{x+y-x}) ÷ d^{y } ——————( applying the rule **a ^{m} ÷ a^{n} = a^{m ― n}**)

= d^{y-y} = d^{0 }= **1**

**Rule 4:** **(ab) ^{n} = a^{n}b^{n}**

**Example**: The simplified value of 2^{2} × 7^{2} × y^{2} = ?

*Solution:*

2^{2} × 7^{2} × y^{2} = (2 × 7 × y)^{2} = (14y)^{2} = **196y**^{2}

**Rule 5: (a ^{n})^{m} = a^{nm}**

**Example:** The value of [ { (2)^{2 }}^{-2} ]^{-1} is equal to:

*Solution*: [ { (2)^{2 }}^{-2} ]^{-1}^{ }= { (2)^{2 }}^{-2}^{×-1} = { (2)^{2} }^{2} = 2^{2×2 }= 2^{4 }= **16**

**Rule 8:** If** a ^{m} = a^{n}** then

**m = n**& if

**a**then

^{m}= b^{m }**a = b.**

**Example:** Find the value of x, if 3^{x+1} + 3^{x} = 36.

*Solution*:

3^{x+1} + 3^{x} = 36.

3^{x} × 3 + 3^{x} = 36

3^{x} × (3 + 1) = 36

3^{x} = 9

3^{x} = 3^{2} ————–(applying the rule **a ^{m} = a^{n}** then

**m = n**)

∴ The value of **x = 2**

Those who are completely new to this topic, it’s my suggestion that you should practice at least 25-30 questions of varying degree of difficulty. So that you hold a good command on the application of various laws of indices.

**Post Your Answer in Comment Section**

- The value of (256)
^{0.16}× (256)^{0.09}is _____________. - If 5
^{y+2}− 5^{y+1}= 500, then find the value of ‘y’.

Thanks Abhinav. Nice explanation with beautiful style of presentation.

Thank You Struggler. Try the last two questions and post your answer in the comments. All the best.

1) (256)^(0.16+0.09) = 2^(8*1/4)= 2^2 = 4

2) y= 2