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# The Laws of Indices

We write as . The small on the top is the number of times appears. We read it as to the power of The power is and is the base.

Likewise,

## Laws of indices

Suppose we wish to simplify

We can write this as .

But this is the same as

The “” appears 8 times. Therefore

### Multiplying indices

When we are multiplying indices, we add the powers, as long as the base is the same.

For example:

### Dividing indices

When we are dividing indices, we subtract powers, as long as the base is the same.

For example:

For example:

Also,

For example:

### When the power is zero

Except from 0, any number raised to the power of 0 is 1.

For example:

### Negative indices

We can also call a negative power a reciprocal

is the reciprocal of

For example:

## Using the Law of Indices to Simplify Numbers and Equations

We can use the laws of indices to simplify numbers and calculations. For example:

If , then

Simplify:

1.

2.

3.

4.

5.

6.

1.

2.

3.

4.

5.

6.

Simplify