# Indices

Indices are also known as powers, exponents or sometimes even orders. Note that indices is plural and index is singular.

Note that in the following:

$x^a$

a is the power/exponent and x is the base. In English, when a letter is smaller and on the upper right side of the bigger letter, we call it a superscript. In Maths, it is often a power. (What are subscripts and superscripts?).

Bear in mind that when you see an expression such as $2x^3$, this is 2 lots of $x$ cubed. This follows from BIDMAS where we apply powers before multiplication. This confuses many students – they often cube 2x which, of course, gives a different answer of $8x^3$.

Students may find it hard to perform tasks with indices at first, especially in an algebraic setting. If you find that you are struggling, take a step back, try doing the calculations with numbers first.

It can be shown that indices abide by the rules that follow.

## The Laws of Indices

• $x^a\times x^b=x^{a+b}$ – think of multiplying $x^2$ by $x^3$. You can write it out in full as $x\times x\times x\times x\times x$. Hence, add the powers. Note that this is only true if the base is the same and should not be applied to $x^2$ by $y^3$, for instance.
• $x^a\div x^b=x^{a-b}$ – similar to the previous example, however, when you are dividing algebraic terms you should subtract the powers.
• $x^0=1$ – anything to the power of zero is 1. You can see this from the previous bullet point by choosing a and b to be the same number.
• $\left(x^a\right)^b=x^{ab}$ – consider taking $x^2$ to the power of 3, i.e. multiplying by itself 3 times. We have $\left(x^2\right)^3=x^2\times x^2\times x^2=x^6$. It follows that we multiply the powers.
• $x^{-n}=\frac{1}{x^n}$ – this is easy to see if you consider $x^3\div x^5=x^{-2}$ and subtracting the powers, then writing it as a fraction: $\frac{x^3}{x^5}=\frac{1}{x^2}$.
• $x^{\frac{1}{n}}=\sqrt[n]{x}$ – can be seen if you consider $x^{\frac{1}{2}}\times x^{\frac{1}{2}}=x$ and so $x^{\frac{1}{2}}$ must be the square root of $x$. This is because $x$ was the result when multiplying something by itself. Multiplying $x^{\frac{1}{3}}$ by itself 3 times shows that $x^{\frac{1}{3}}=\sqrt{x}$ and the same applies for other fractions.  It follows from this rule that $x^{\frac{m}{n}}=\left(\sqrt[n]{x}\right)^m$.

See Examples.

## Indices Examples

• $2^7\times 2^9=2^{16}$
• $\frac{4x^7}{2x^3}=2x^4$
• $\left(3p^2\right)^4=3^4(p^2)^4=81p^8$
• $16\times 2^{-3}=16\times \frac{1}{2^3}=16\times \frac{1}{8}=2$

$\left(\frac{8}{27}\right)^{\frac{2}{3}}=\left(\left(\frac{8}{27}\right)^{\frac{1}{3}}\right)^2=\left(\sqrt{\frac{8}{27}}\right)^2=\left(\frac{2}{3}\right)^2=\frac{4}{9}$