Indices show where a number has been multiplied by itself, eg squared or cubed, or to show roots of numbers, eg square root. Some terms with indices can be simplified using the laws of indices.

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The two basic laws of indices are:

\[{a^m} \times {a^n} = {a^{m + n}}\]

\[{a^m} \div {a^n} = {a^{m - n}}\]

Try to use these to work through the example questions below.

- Question
Simplify \({y^7} \times {y^3} \times {y^5}\)

Use the multiplication law. This tells you to add the indices.

\[= {y^{7 + 3 + 5}} = {y^{15}}\]

- Question
Simplify \({y^{10}} \div {y^3}\)

This could also have been written as:

\[\frac{{{y^{10}}}}{{{y^3}}}\]

Use the division law which tells you to subtract the indices.

\[= {y^{10 - 3}} = {y^7}\]

- Question
Simplify \(\frac{{{y^7} \times {y^4}}}{{{y^5}}}\)

\[= \frac{{{y^{7 + 4}}}}{{{y^5}}}\]

Use the multiplication law, add the numerator indices.

\[= \frac{{{y^{11}}}}{{{y^5}}}\]

Use the division law, subtract the indices.

\[= {y^{11 - 5}} = {y^6}\]

- Question
Simplify \(y \times {y^8} \times {y^4}\)

\[y \times {y^8} \times {y^4}\]

\[=y^{1+8+4}\]

Remember \(y = {y^1}\)

\[=y^{13}\]

- Question
Simplify \({y^6} \times {y^0}\)

\[= {y^{6 + 0}} = {y^6}\]

This shows that \({y^0} = 1\)