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

Algebra ->  Exponents-negative-and-fractional -> SOLUTION: Find all solution of (x^2+9)^3/2=125 I appreciate.      Log On


   

Question 355081: Find all solution of (x^2+9)^3/2=125 I appreciate.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
%28x%5E2%2B9%29%5E%283%2F2%29=125
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, %28a%5Ep%29%5Eq+=+a%5E%28%28p%2Aq%29%29, 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:
%28%28x%5E2%2B9%29%5E%283%2F2%29%29%5E%282%2F3%29=%28125%29%5E%282%2F3%29
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 125+=+5%5E3, I'm going to find the cube root first:
%28%28x%5E2%2B9%29%5E%283%2F2%29%29%2A%282%2F3%29=%28125%29%5E%282%2A%281%2F3%29%29
%28x%5E2%2B9%29%5E1=%28125%5E%281%2F3%29%29%5E2
x%5E2%2B9=5%5E2
x%5E2%2B9=25
Now we have a quadratic equation to solve. Subtract 25 from each side:
x%5E2+-+16+=+0
Factor the difference of squares:
%28x+%2B+4%29%28x+-+4%29+=+0
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