SOLUTION: Simplify each of the following expressions. Write your answers with positive exponents only. (x^2)^-3(x^-2) / (x^2)^-4

Algebra ->  Expressions -> SOLUTION: Simplify each of the following expressions. Write your answers with positive exponents only. (x^2)^-3(x^-2) / (x^2)^-4       Log On


   

Question 77227: Simplify each of the following expressions. Write your answers with positive exponents only.
(x^2)^-3(x^-2) / (x^2)^-4
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Given
.
%28x%5E2%29%5E%28-3%29%2A%28x%5E-2%29%2F%28x%5E2%29%5E%28-4%29
.
You can apply the power rule of exponents to terms of this expression. When a base quantity
that has an exponent is then raised to a power, it is equivalent to the base quantity
raised to the product of the two exponents.
.
For example, the first term in the numerator is:
.
%28x%5E2%29%5E%28-3%29
.
The product of the two exponents is %28-3%29%2A2+=+-6 and as a result, the equivalent
term is:
.
x+%5E+%28-6%29
.
Applying the same technique to the denominator results in the denominator becoming:
.
%28x%5E2%29%5E%28-4%29=+x%5E%282%2A%28-4%29%29+=+x%5E%28-8%29
.
Making these two substitutions into the expression results in:
.
x%5E%28-6%29%2Ax%5E%28-2%29%2Fx%5E%28-8%29
.
Notice that the terms all have x as the base. Therefore they can be multiplied by just
adding the exponents and divided by subtracting the exponents. First apply this rule to
the two terms in the numerator to get:
.

.
Note that the numerator is the same as the denominator. When they are divided the result
is obtained by subtracting the exponent in the denominator from the exponent in the
numerator to get x%5E%28-8+-+%28-8%29%29+=+x%5E%28-8%2B8%29=+x%5E0 and when any quantity is raised to the exponent
zero, the result is 1.
.
So, the simplification of this problem results in an answer of 1.
.
Hope this helps you to understand the problem. There are other ways to manipulate
the exponents, but they also result in the same answer ... the expression simplifies
to an answer of 1.