# How do you simplify y^9÷y^5?

Jan 21, 2016

${y}^{9} \div {y}^{5} = {y}^{4}$

#### Explanation:

${y}^{9} = {y}^{5} \times {y}^{4}$

Therefore
$\textcolor{w h i t e}{\text{XXX}} {y}^{9} \div {y}^{5} = \frac{{y}^{9}}{{y}^{5}} = \frac{\cancel{{y}^{5}} \times {y}^{4}}{\cancel{{y}^{5}}} = {y}^{4}$

Jan 21, 2016

So the expression simplifies to ${y}^{4}$

#### Explanation:

$\textcolor{b l u e}{\text{An introduction to the idea of powers (indices).}}$

Consider the variable $y$. This can be written as ${y}^{1}$.

Now consider ${y}^{1} \times {y}^{1}$ in the same way you would think about ${2}^{1} \times {2}^{1}$. It is accepted that ${2}^{1} \times {2}^{1}$can be written as ${2}^{2}$.

This the same as ${2}^{1 + 1}$.
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Now consider ${y}^{1} \times {y}^{1} \times {y}^{1}$ in the same way that ${2}^{1} \times {2}^{1} \times {2}^{1} = {2}^{1 + 1 + 1} = {2}^{3} '$

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Now think about ${2}^{5}$. By what I wrote previously this can be split into, say; ${2}^{3 + 2}$. It is still has the same value as ${2}^{5}$ but looks different.
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$\textcolor{b l u e}{\text{Answering your question}}$

$\textcolor{b r o w n}{\text{Given: } {y}^{9} \div {y}^{5}}$

Write this as: $\frac{{y}^{9}}{{y}^{5}}$.............................(1)

$\textcolor{p u r p \le}{\text{But "y^9" may be written as } {y}^{5 + 4}}$

$\textcolor{p u r p \le}{\text{and } {y}^{5 + 4} = {y}^{5} \times {y}^{4}}$.....................(2)

$\textcolor{b r o w n}{\text{Substitute Expression (2) into expression (1)}}$

$\frac{{y}^{9}}{{y}^{5}} \to \frac{{y}^{5} \times {y}^{4}}{{y}^{5}}$

This the same as: ${y}^{5} / {y}^{5} \times {y}^{4}$

But ${y}^{5} / {y}^{5} = 1$ giving:

$1 \times {y}^{4}$

So the expression simplifies to ${y}^{4}$

Jan 21, 2016

${y}^{4}$

#### Explanation:

Just a more extended way of saying what's already been said:

${y}^{9} / {y}^{5} = \frac{y \times y \times y \times y \times \textcolor{red}{\cancel{\textcolor{b l a c k}{y}}} \times \textcolor{red}{\cancel{\textcolor{b l a c k}{y}}} \times \textcolor{red}{\cancel{\textcolor{b l a c k}{y}}} \times \textcolor{red}{\cancel{\textcolor{b l a c k}{y}}} \times \textcolor{red}{\cancel{\textcolor{b l a c k}{y}}}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{y}}} \times \textcolor{red}{\cancel{\textcolor{b l a c k}{y}}} \times \textcolor{red}{\cancel{\textcolor{b l a c k}{y}}} \times \textcolor{red}{\cancel{\textcolor{b l a c k}{y}}} \times \textcolor{red}{\cancel{\textcolor{b l a c k}{y}}}} = y \times y \times y \times y = {y}^{4}$

Jan 24, 2017

${y}^{4}$
${a}^{n} \div {a}^{m} = {a}^{n - m}$
$\therefore {y}^{9} \div {y}^{5} = {y}^{9 - 5} = {y}^{4}$