Posted by: catindiaonline | October 25, 2008

## Indices

As you should know from GCSE, x3 is a shortening of $x \times x \times x$. In the same way, any number to the power of n is that number multiplied by itself n times. To describe more detail, in the expression x3, the x is referred to as the base, and the 3 as the exponent.

By always reducing the expression back to its basic parts it is possible to come to conclusions about how to treat numbers raised to powers in algebra.

### Multiplication

When you multiply indices you add the exponents together.

As you can see $x^3 \times x^2 = x^{3+2} = x^5$ .

### Division

Division is, as expected, the opposite to multiplication. When you divide indices you subtract the exponents from each other.

$\frac{x^4}{x^2} = \frac{x \times x \times x \times x}{x \times x} = \frac{\not \! x \times \not \! x \times x\times x}{\not \! x \times \not \! x} = x \times x = x^2$

Again this shows that $\frac{x^4}{x^2} = x^{4-2} = x^2$

### Negative powers

Having done the above division, we begin to ask ourselves what happens when we have a fraction that has a higher exponent on the bottom than the top.

Using the trusted method of showing each x seperately we obtain:

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