Algebra and calculus

Laws of indices

Algebra uses symbols or letters to represent quantities; for example I = PRT

I is used to stand for interest, P for principle, R for rate, and T for time.

A quantity made up of symbols together with operations ( $+ - \times \div$ ) is called an algebraic expression. We use the laws of indices to simplify expressions involving indices.

Expand the following boxes for the laws of indices. The videos show why the laws are true.

The first law: multiplication

If the two terms have the same base (in this case $x$ ) and are to be multiplied together their indices are added.

In general: $x^m \times x^n = x^{m+n}$

Here is an explanation of why this is true:

The second law: division

If the two terms have the same base (in this case $x$ ) and are to be divided their indices are subtracted.

In general: $\dfrac{x^m}{x^n}=x^{m-n}$

Here is an explanation of why this is true:

The third law: brackets

If a term with a power is itself raised to a power then the powers are multiplied together.

In general: $(x^m)^n = x^{m \times n}$

Here is an explanation of why this is true:

Negative powers

Consider this example: $\dfrac{a^2}{a^6} = a^{2-6} = a^{-4}$

Also we can show that: $\dfrac{a^2}{a^6} = \dfrac{1}{a^4}$

So a negative power can be written as a fraction.

In general: $x^{-m} = \dfrac{1}{x^m}$

Here is an explanation of why this is true:

Power of zero

The second law of indices helps to explain why anything to the power of zero is equal to one.

We know that anything divided by itself is equal to one. So $\dfrac {x^3}{x^3} = 1$

Also we know that $\dfrac {x^3}{x^3} = x^{3-3} = x^0 = 1$

Therefore, we have shown that $\dfrac {x^3}{x^3} = x^0 = 1$

Here is an explanation of why this is true:

Fractional powers

Both the numerator and denominator of a fractional power have meaning.

The bottom of the fraction stands for the type of root; for example, $x^{\frac{1}{3}}$ denotes a cube root $\sqrt[3]{x}$

The top line of the fractional power gives the usual power of the whole term.

For example: $x^{\frac{2}{3}} = (\sqrt[3]{x})^2$

In general: $x^{\frac{m}{n}} = (\sqrt[n]{x})^m$

This explanation show why a root is shown as a fractional power:

Further information

Press the Printer Friendly button at the bottom left-hand corner to download a printable handout
RMIT University has online videos to review the use of indices including fractional indices, brackets and logarithms.

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