# How do you evaluate the expression (2/3)^4/((2/3)^-5(2/3)^0) using the properties of indices?.

May 23, 2017

${\left(\frac{2}{3}\right)}^{4} / \left({\left(\frac{2}{3}\right)}^{-} 5 {\left(\frac{2}{3}\right)}^{0}\right) = {\left(\frac{2}{3}\right)}^{9} = \frac{512}{19683}$

#### Explanation:

We can use here the identities

${a}^{m} \times {a}^{n} = {a}^{\left(m + n\right)}$ and ${a}^{m} / {a}^{n} = {a}^{\left(m - n\right)}$

As such ${a}^{m} / \left({a}^{n} {a}^{p}\right) = {a}^{\left(m - n - p\right)}$

Hence ${\left(\frac{2}{3}\right)}^{4} / \left({\left(\frac{2}{3}\right)}^{-} 5 {\left(\frac{2}{3}\right)}^{0}\right)$

$= {\left(\frac{2}{3}\right)}^{\left(4 - \left(- 5\right) - 0\right)}$

$= {\left(\frac{2}{3}\right)}^{\left(4 + 5\right)}$

$= {\left(\frac{2}{3}\right)}^{9}$

or ${2}^{9} / {3}^{9} = \frac{512}{19683}$

May 23, 2017

${2}^{9} / {3}^{9} = {\left(\frac{2}{3}\right)}^{9}$

$= \frac{512}{19683}$

#### Explanation:

There are four properties of indices (exponents) to consider here:

Raising factors to a power: $\textcolor{b l u e}{{\left(x y\right)}^{m} = {x}^{m} \times {y}^{m}}$

A negative index: $\textcolor{m a \ge n t a}{\frac{1}{x} ^ - m = {x}^{m}}$

Index of $0$. color(lime)("Anything to power of 0 is equal to"1)" "(except ${0}^{0}$)

Multiply law: same bases, add the indices: ${x}^{m} \times {x}^{n} = {x}^{m + n}$

color(blue)((2/3)^4)/(color(magenta)((2/3)^-5)color(lime)((2/3)^0)) = (color(blue)(2^4/3^4)xxcolor(magenta)((2/3)^5))/(color(lime)((1))

$= {2}^{4} / {3}^{4} \times {2}^{5} / {3}^{5}$

$= {2}^{9} / {3}^{9}$

This can also be written as ${\left(\frac{2}{3}\right)}^{9}$

$= \frac{512}{19683}$