SOLUTION: Find all solution of (x^2+9)^3/2=125 I appreciate.

Question 355081: Find all solution of (x^2+9)^3/2=125 I appreciate.
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

First of all we need to get rid of that exponent of 3/2. How do you make an exponent "disappear"? When do exponents become "invisible"? The answer to both questions is: when the exponent becomes a 1!

So we need to think about how we can use proper Math to change an exponent of 3/2 into a 1. We should know that whenever you multiply reciprocals you get a 1. And you should know that there is a rule for exponents, , which tels you when it is correct to multiply exponents. Putting these together, we find that raising the 3/2 power to the 2/3 (2/3 is the resiprocal of 3/2) power then the exponent will become a 1 and "disappear"! So we will raise both sides of this equation to the 2/3 power:

Simplifying on the left is easy since both exponents combine to form a 1 which "disappears". On the right side we want to raise 125 to the 2/3 power. In the exponent of 2/3 the 2 means we will square something and the 1/3 means we will find a cube root of something. And we can do these in any order. Since , I'm going to find the cube root first:

Now we have a quadratic equation to solve. Subtract 25 from each side:

Factor the difference of squares:

From the Zero Product Property we know that this (or any) product can be zero only if one of the factors is zero. So:
x + 4 = 0 or x - 4 = 0
Solving these we get:
x = -4 or x = 4
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