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 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)#
