Question #3c8d4

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Answer 1 · 2015-10-15T00:22:48

$n = 22/15$

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 $n$

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

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

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,

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