# 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 $${y^0} = 1$$