# Question #a6e54

Jan 27, 2017

Expression $\textcolor{red}{D}$ has the greatest value for $z = 12$

#### Explanation:

First, we need to simplify each of the expression using rules for exponents and then evaluate with $z = 12$:

$\textcolor{red}{A}$) Use these rules for exponents:
${x}^{\textcolor{red}{a}} \times {x}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} + \textcolor{b l u e}{b}}$
${x}^{\textcolor{red}{a}} = \frac{1}{x} ^ \textcolor{red}{- a}$

${z}^{-} 6 {z}^{4} = {z}^{- 6 + 4} = {z}^{-} 2 = \frac{1}{z} ^ \left(- - 2\right) = \frac{1}{z} ^ 2$

$\frac{1}{12} ^ 2 = \frac{1}{144} = 0.00694$

$\textcolor{red}{B}$) Use these rules for exponents:
${x}^{\textcolor{red}{a}} \times {x}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} + \textcolor{b l u e}{b}}$
${x}^{\textcolor{red}{a}} = \frac{1}{x} ^ \textcolor{red}{- a}$
${\left({x}^{\textcolor{red}{a}}\right)}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} \times \textcolor{b l u e}{b}}$

${\left({z}^{-} 2 {z}^{5}\right)}^{-} 2 = {\left({z}^{- 2 + 5}\right)}^{-} 2 = {\left({z}^{3}\right)}^{-} 2 = {z}^{3 \times - 2} = {z}^{-} 6 = \frac{1}{z} ^ 6$

$\frac{1}{12} ^ 6 = \frac{1}{2985984} = 3.35 x {10}^{-} 7 = 0.000000335$

$\textcolor{red}{C}$) Use these rules for exponents:
${\left({x}^{\textcolor{red}{a}}\right)}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} \times \textcolor{b l u e}{b}}$

${\left({z}^{3}\right)}^{5} = {z}^{3 \times 5} = {z}^{15}$

${12}^{15} = 1.54 x {10}^{16} = 15 , 400 , 000 , 000 , 000 , 000$

$\textcolor{red}{D}$) Use these rules for exponents:
${x}^{\textcolor{red}{a}} \times {x}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} + \textcolor{b l u e}{b}}$
${\left({x}^{\textcolor{red}{a}}\right)}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} \times \textcolor{b l u e}{b}}$

$- {\left({z}^{2} {z}^{-} 4\right)}^{-} 3 = - {\left({z}^{2 - 4}\right)}^{-} 3 = - {\left({z}^{-} 2\right)}^{-} 3 = - {z}^{- 2 \times - 3} = - {z}^{6}$

$- {\left(12\right)}^{6} = - 2 , 985 , 984$