# How do you evaluate 96^2*(1/3^2)^3*6^2?

Apr 30, 2016

$\frac{4096}{9}$

#### Explanation:

Given,

${96}^{2} \cdot {\left(\frac{1}{3} ^ 2\right)}^{3} \cdot {6}^{2}$

Break down the first base into prime numbers.

$= {\left({2}^{5} \cdot 3\right)}^{2} \cdot {\left(\frac{1}{3} ^ 2\right)}^{3} \cdot {6}^{2}$

Simplify.

$= {2}^{10} \cdot {3}^{2} \cdot \frac{1}{3} ^ 6 \cdot {6}^{2}$

$= {2}^{10} \cdot {3}^{2} / {3}^{6} \cdot {6}^{2}$

$= {2}^{10} \cdot {3}^{2} / \left({3}^{2} \cdot {3}^{4}\right) \cdot {6}^{2}$

$= {2}^{10} \cdot \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{{3}^{2}}}}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{{3}^{2}}}} \cdot {3}^{4}} \cdot {6}^{2}$

Break down the last base into prime numbers.

$= {2}^{10} \cdot \frac{1}{3} ^ 4 \cdot {6}^{2}$

$= {2}^{10} \cdot \frac{1}{3} ^ 4 \cdot {\left(2 \cdot 3\right)}^{2}$

$= {2}^{10} \cdot \frac{1}{{3}^{2} \cdot {3}^{2}} \cdot {2}^{2} \cdot {3}^{2}$

$= {2}^{10} \cdot \frac{1}{{3}^{2} \cdot \textcolor{red}{\cancel{\textcolor{b l a c k}{{3}^{2}}}}} \cdot {2}^{2} \cdot \textcolor{red}{\cancel{\textcolor{b l a c k}{{3}^{2}}}}$

$= {2}^{10} \cdot \frac{1}{3} ^ 2 \cdot {2}^{2}$

$= 1024 \cdot \frac{1}{9} \cdot 4$

$= \frac{4096}{9}$

May 1, 2016

The answer can be left in index form. This has more meaning than the actual numbers. ${2}^{12} / {3}^{2}$

#### Explanation:

A quicker method would be to work with all the bases at the same time. Change any base to prime factors.

$= {\left({2}^{5} \cdot 3\right)}^{2} \cdot {\left(\frac{1}{3} ^ 2\right)}^{3} \cdot {\left(2 \cdot 3\right)}^{2}$

Simplify by removing the brackets.

$= {2}^{10} \cdot {3}^{2} \cdot \frac{1}{3} ^ 6 \cdot {2}^{2} \cdot {3}^{2}$

Combine like bases by adding the indices:

$= {2}^{12} \cdot {3}^{4} / {3}^{6}$

Finally subtract the indices of like bases

$= {2}^{12} / {3}^{2}$