# Evaluate 4(1/3)^7 div 2(1/3)^3 ?

Dec 6, 2017

The expression simplifies to $\frac{2}{81}$

#### Explanation:

It looks like the $4$ and the $2$ are coefficients.

$4 {\left(\frac{1}{3}\right)}^{7} \div 2 {\left(\frac{1}{3}\right)}^{3}$

1) Using a calculator, evaluate the fractions raised to the power of $7$ and the power of $3$

$4 \left(\frac{1}{2187}\right) \div 2 \left(\frac{1}{27}\right)$

2) Clear the parentheses by distributing the coefficients

$\frac{4}{2187} \div \frac{2}{27}$

3) Divide the fractions by multiplying by the reciprocal of the divisor

$\frac{4}{2187} \times \frac{27}{2}$

4) Reduce the fraction to lowest terms

${\cancel{4}}^{2} / {\cancel{2187}}^{81} \times {\cancel{27}}^{1} / {\cancel{2}}^{1}$

5) The fractions reduce to

$\frac{2}{81} \times \frac{1}{1}$

$\frac{2}{81}$

Dec 6, 2017

$\frac{2}{81}$

#### Explanation:

Expression $= 4 {\left(\frac{1}{3}\right)}^{7} \div 2 {\left(\frac{1}{3}\right)}^{3}$

To make this clearer, let's rewrite the expression as follows:

Expression$= \frac{4 \times {\left(\frac{1}{3}\right)}^{7}}{2 \times {\left(\frac{1}{3}\right)}^{3}} = 2 \times \frac{{\left(\frac{1}{3}\right)}^{7}}{{\left(\frac{1}{3}\right)}^{3}}$

Now we can use the law of exponents that states:

$\frac{{a}^{m}}{{a}^{n}} = {a}^{m - n}$

Thus, Expression $= 2 \times {\left(\frac{1}{3}\right)}^{7 - 3}$

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

$= \frac{2}{3} ^ 4 = \frac{2}{81}$