# Question #3c8d4

Oct 15, 2015

$n = \frac{22}{15}$

#### Explanation:

Your starting equation looks like this

${\left({a}^{4}\right)}^{\frac{1}{5}} \cdot {\left({a}^{2}\right)}^{\frac{1}{3}} = {a}^{n}$

You will use two properties of exponents to find the value of $n$

• the power of a power property
• power of a product property

Let's start with the first one. According to the power of a power property of exponents, you have

$\textcolor{b l u e}{{\left({a}^{x}\right)}^{y} = {a}^{x \cdot y}}$

${\left({a}^{4}\right)}^{\frac{1}{5}} = {a}^{4 \cdot \frac{1}{5}} = {a}^{\frac{4}{5}}$

and

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

So the equation becomes

${a}^{\frac{4}{5}} \cdot {a}^{\frac{2}{3}} = {a}^{n}$

Now focus on the second property, the power of a product property, which tells you that

$\textcolor{b l u e}{{a}^{x} \cdot {a}^{y} = {a}^{x + y}}$

${a}^{\frac{4}{5}} \cdot {a}^{\frac{2}{3}} = {a}^{\frac{4}{5} + \frac{2}{3}} = {a}^{\frac{4 \cdot 3 + 2 \cdot 5}{3 \cdot 5}} = {a}^{\frac{22}{15}}$
${a}^{\frac{22}{15}} = {a}^{n}$
${a}^{\frac{22}{15}} = {a}^{n} \iff n = \textcolor{g r e e n}{\frac{22}{15}}$