# Question #3c8d4

##### 1 Answer

#### Explanation:

Your starting equation looks like this

#(a^4)^(1/5) * (a^2)^(1/3) = a^n#

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

the power of a power propertypower of a product property

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

#color(blue)((a^x)^y = a ^(x * y))#

In your case, you have

#(a^4)^(1/5) = a^(4 * 1/5) = a^(4/5)#

and

#(a^2)^(1/3) = a^(2 * 1/3) = a^(2/3)#

So the equation becomes

#a^(4/5) * a^(2/3) = a^n#

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

#color(blue)(a^(x) * a^(y) = a^(x+y))#

In your case, you have

#a^(4/5) * a^(2/3) = a^(4/5 + 2/3) = a^((4 * 3 + 2 * 5)/(3 * 5)) = a^(22/15)#

The equation is now

#a^(22/15) = a^n#

In order for these two terms to be equal, you need the *exponents* of the **equal**. In other words,

#a^(22/15) = a^(n) <=> n = color(green)(22/15)#