# Index laws

**Multiplying numbers in index form**

When we multiply two numbers written in index form and with the same base we can see an interesting result.

For example:

3⁴ x 3² = (3 x 3 x 3 x 3) x (3 x 3)

= 3 x 3 x 3 x 3 x 3 x 3

= 3⁶

= 3(⁴+²)

7³ x 7⁵ = (7 x 7 x 7) x (7 x 7 x 7 x 7 x 7)

= 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x

= 7⁸

= 7(³+⁵)

Therefore when we multiply two numbers with the **same base**, the indices are added.

**Dividing numbers in index form**

When we divide two numbers written in index form and with the same base we can see another interesting result.

For example:

Therefore when we divide two numbers with the **same base** the indices are subtracted.

**Raising a power to a power**

Numbers can be raised to a power and the result raised to another power.

For example,

(4³)² = 4³ x 4³

= (4 x 4 x 4) x (4 x 4 x 4)

= 4⁶

= 4(³ x ²)

Therefore, when a number of the same base is raised to a power and then raised to another power, the powers are multiplied.

**The power of 1**

Any number raised to the power of 1 is equal to its original base number.

**The power of 0**

Any non-zero number raised to the power of 0 is equal to 1. Also note that 0 cannot be the power of 0! (If you enter this on a calculator, an error will be displayed).