<s>[INST] Need Help Finding Solution to These 5 Simple Exponent Exercises In this document, I have 5 exercises involving exponents and their five corresponding solutions. I am trying to find out why these are the solutions for these equations, as I cannot seem to get the same results as the provided solutions. The exercises ask to simplify the following equations. Can anyone assist in explaining how to solve for these five equations? 
Thank you!  
Exponent Equations
http://i349.photobucket.com/albums/q368/Chaplive/Screenshot%2035_zpsx2sjpp5f.png
 [/INST] $1)$ Use the law of exponents ie $a^{-n}=\frac1{a^n}$
so $(-3)^4\div3^5=\frac{(-1)^43^4}{3^5}=3^{4-5}=3^{-1}$
$2)$ $\dfrac{2^{-1}+2^{-2}}{2^{-1}\cdot2^{-2}}=\frac{2^{-1}}{2^{-3}}+\frac{2^{-2}}{2^{-3}}= 2^2+2=6$
$3)$ note that $(a\cdot b)^c = a^c\cdot b^c$
$\dfrac{81^4\cdot4^9}{6^{16}}= \dfrac{3^{4\cdot4}\cdot2^{2\cdot9}}{3^{16}\cdot2^{16}}=\dfrac{3^{16}\cdot2^{18}}{3^{16}\cdot2^{16}}= 2^2 =4$
$4)$ remember that $(-1)^{even}= 1$ and $(-1)^{odd}=-1$
$-(-1)^2-(-1^5)-(-1)^7-(-1)^{100} = -(1)-(-1)-(-1)-(1) =-1+1+1-1 = 0$
$5)$ $a^0 =1$  so,
$\dfrac{(5^{-1}+5^{-2})^0}{5^{1}+5^{-2}}= \dfrac{1}{\frac15+\frac1{25}}=\frac{1}{\frac{5+1}{25}}=\dfrac{25}6$
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