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Simplifying Irrational Expressions by Christopher Monahan

Recall two important properties of exponents: (ab)n = an bn (when a product of terms is raised to a power, the answer is equal to the product of the terms each raised to a power) and

(when a quotient of terms is raised to a power, the answer is equal to the quotient of the terms each raised to a power).

A consequence of these rules is that when the exponents are fractions,

Because this is a square root function, you want to consider those factors of 8 that are squares; 4 and 2 work. By the property of exponents,

becomes
.

When simplifying

, you must remove all factors of x that are perfect nth powers.

Simplify:

432 is a relatively big number, and finding the largest perfect square factor might not be something you can do. You do not have to get the largest factor the first time; you just need to be sure that your final answer does not have any perfect square factors (other than 1, of course). Because 4 + 3 + 2 = 9, we know that 9 is a factor of 432. 432 = 9 * 48. 4 is a factor of 48, so 432 = 9 * 4 * 12. And 4 is also a factor of 12, so 432 = 9 * 4 * 4 * 3. Therefore,

.

Simplify:

You are looking for perfect cube factors of 108 because the index is a 3. 108 = 27 * 4, so

.

Simplify

. Assume the variables represent positive numbers.

Reduce the fraction to get

. Because each of the terms is a perfect square, the result contains no square roots.
.

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