How do you simplify #y^9÷y^5#?

#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#

#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)#.
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Now consider #y^1xxy^1xxy^1# in the same way that #2^1xx2^1xx2^1=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.
#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#

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#

Exponents quotient rule:

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

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