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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:    ### The rules for brackets 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    ### Example

Simplify:

1. 2. 3. 4. 5. 6. #### Solutions:

1. 2. 3. 4. 5. 6. ### Example

Simplify #### Solution:    