Indices

Simplifying indices

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 \({y^0} = 1\)