Question

How do you simplify `y^9÷y^5`?

Answer 1

`y^9 div y^5 = y^4`

Explanation:

`y^9 = y^5xxy^4`

Therefore
`color(white)("XXX")y^9 div y^5 = (y^9)/(y^5) = (cancel(y^5)xxy^4)/(cancel(y^5)) = y^4`

Answer 2

So the expression simplifies to `y^4`

Explanation:

`color(blue)("An introduction to the idea of powers (indices).")`

Consider the variable `y`. This can be written as `y^1`.

Now consider `y^1xxy^1` in the same way you would think about `2^1xx2^1`. It is accepted that `2^1xx2^1`can be written as `2^2`.

This the same as `2^(1+1)`.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Now consider `y^1xxy^1xxy^1` in the same way that `2^1xx2^1xx2^1=2^(1+1+1)=2^3'`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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.
`color(blue)("++++++++++++++++++++++++++++++++++++++")`

`color(blue)("Answering your question")`

`color(brown)("Given: "y^9-:y^5)`

Write this as: `(y^9)/(y^5)`.............................(1)

`color(purple)("But "y^9" may be written as "y^(5+4))`

`color(purple)("and "y^(5+4) = y^5xxy^4)`.....................(2)

`color(brown)("Substitute Expression (2) into expression (1)")`

`(y^9)/(y^5)->(y^5xxy^4)/(y^5)`

This the same as: `y^5/y^5 xxy^4`

But `y^5/y^5=1` giving:

`1xxy^4`

So the expression simplifies to `y^4`

Answer 3

`y^4`

Explanation:

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

`y^9/y^5=(yxxyxxyxxyxxcolor(red)(cancel(color(black)(y)))xxcolor(red)(cancel(color(black)(y)))xxcolor(red)(cancel(color(black)(y)))xxcolor(red)(cancel(color(black)(y)))xxcolor(red)(cancel(color(black)(y))))/(color(red)(cancel(color(black)(y)))xxcolor(red)(cancel(color(black)(y)))xxcolor(red)(cancel(color(black)(y)))xxcolor(red)(cancel(color(black)(y)))xxcolor(red)(cancel(color(black)(y))))=yxxyxxyxxy=y^4`

Answer 4

`y^4`

Explanation:

Exponents quotient rule:

`a^n-:a^m=a^(n-m)`

`:.y^9-:y^5=y^(9-5)=y^4`