SOLUTION: simplify (25/x^-4)^(3/2)

Question 398745: simplify (25/x^-4)^(3/2)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

First let's work within the parentheses. In general and . Using the second pattern, the expression within the parentheses becomes:

Now we can use a rule for exponents, , to raise to the 3/2 power:

Simplifying the last part of the expression is simple. The rule for exponents when raising a power to a power is to multiply the exponents:

Simplifying the first part can be difficult for those who do not yet understand fractional exponents. I find that it can help if you factor the exponent in a special way:

If the exponent is negative, factor out -1.
If the exponent is fractional and the numerator is not a 1, then factor out the numerator.

The exponent on the 25 is not negative. But it is fractional and its denominator is not a 1. So we will factor out the 3:

Each of the factors tells us an operation to perform. The 3 tells us that we will be cubing. And the 1/2, if you remember what fractional exponents mean, tells us that we will be finding a square root. And since multiplication is Commutative, we can do these operations in any order we choose. Cubing 25 doesn't look very appealing while finding a square root of 25 looks very easy. So we will start with the square root:

Note: If we had chosen to cube 25 first and then find a square root, we would still have gotten 125 -- just not as easily.
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