When you have a negative exponent, you move the base with the negative exponent to the other side of the fraction to make the exponent positive.
For example:
[tex]\frac{1}{y^{-2}} =\frac{y^2}{1}[/tex] or [tex]y^2[/tex]
[tex]2x^{-3}[/tex] or [tex]\frac{2^1x^{-3}}{1} =\frac{2}{x^3}[/tex]
[tex]\frac{1}{x^{-5}} =x^5[/tex]
When a base with an exponent is divided by a base with an exponent, you subtract the exponents together. (But you can only combine the exponents when the bases are the same)
For example:
[tex]\frac{x^2}{y}[/tex] (can't combine because they have different bases of y and x)
[tex]\frac{x^5}{x^3} =x^{(5-3)}=x^2[/tex]
[tex]\frac{2^2}{2} =2^{(2-1)}=2^1 = 2[/tex]
When you multiply an exponent directly to a base with an exponent, you multiply the exponents together.
For example:
[tex](x^2)^3=x^{(2*3)}=x^6[/tex]
[tex](y^2)^{10}=y^{(2*10)}=y^{20}[/tex]
[tex]\frac{2e^0}{(e^{-3})^2}[/tex] First multiply the exponents together in the denominator
[tex]\frac{2e^0}{e^{(-3*2)}}= \frac{2e^0}{e^{-6}}[/tex] Now subtract the exponents together
[tex]2e^{(0-(-6))}[/tex] (two negative signs cancel each other out and become positive)
[tex]2e^{(0+6)}[/tex]
[tex]2e^6[/tex]
