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.

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 powers are applied before multiplication. Many students get this confused with cubing 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 following rules in maths:

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, the powers are added. 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 the powers are multipled.
  • 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 something multiplied by itself made x. Taking x^{\frac{1}{3}} multiplied by itself 3 times shows that x^{\frac{1}{3}}=\sqrt[3]{x} and the same applies for other fractions.  It follows from this rule that x^{\frac{m}{n}}=\left(\sqrt[n]{x}\right)^m.

Click here to see this list on the Things to Remember page.


Example 1

  • 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

Example 2

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


Alternatively, click here to find Questions by Topic and scroll down to all past INDICES questions to practice some more.


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