Roots and rational exponents

Raise both sides to the reciprocal of the original exponent. This is valid for positive x. Check the resulting value in the original equality.

Add exponents when multiplying powers with the same base, then multiply exponents when raising a power to a power. Calculate fractional exponents exactly and reduce the result.

Extract square factors and cancel common radicals. Add the resulting fractions using a common denominator.

When multiplying two roots with the same index, their radicands may be multiplied under one radical:

Rule: ⁿ√a · ⁿ√b = ⁿ√(a · b).

1. Check that both roots have the same index n.
2. Identify the numbers whose nth powers give the first and second radicands.
3. Evaluate both roots and multiply the resulting numbers.

In this question, both radicands are generated as exact nth powers, so the answer is an integer.

Extract square factors from both roots. Factor the common root out of the difference, square the whole difference, then simplify the quotient.

A root taken from another root can be written as one root.

1. Multiply the indices of the outer and inner roots.
2. Take the root with that new index from the original number.
3. Check the answer: raising it to the product of the two indices must give the original number.

Rule: the nth root of the mth root of a number is the (n × m)th root of that number.

A fractional exponent describes two operations: the denominator n gives the root index, while the numerator m gives the power.

Rule: a^m/n = ⁿ√(a^m) = (ⁿ√a)^m.

1. Use the denominator of the fractional exponent as the root index.
2. Keep the numerator as the power of the radicand or of the complete root.
3. Check that m and n have not been interchanged.

Over the real numbers, when n is even, the radicand must be non-negative.

Find two squares whose sum matches the first term and whose product matches the inner radicand. Apply the square-of-a-sum or square-of-a-difference identity. Remember that the square root is non-negative.

First evaluate the positive rational power: the denominator gives the root index, and the numerator gives the power. The negative exponent means taking the reciprocal.

The fractional exponent ^m⁄_n means that you may first take the nth root and then raise the result to the power m.

Rule: a^m/n = (ⁿ√a)^m = ⁿ√(a^m).

1. Look at the denominator n of the exponent: it gives the root index.
2. Take the nth root of the base.
3. Raise the resulting number to the power m shown by the numerator.

In this question, the base is generated as an exact nth power, so the root is an integer.