Roots and rational exponents

Over the real numbers, an even root is defined only when its radicand is non-negative, meaning greater than or equal to zero.

Condition: if n is even, the expression ⁿ√a requires a ≥ 0.

• If a > 0, the root has a positive real value.
• If a = 0, the root equals 0.
• If a < 0, an even root does not exist over the real numbers.

This is because raising any real number to an even power produces a non-negative result.

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.

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.

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.

To rationalise the denominator, multiply the numerator and denominator by √{n}.

1/√{n} · √{n}/√{n} = √{n}/n.

This leaves the rational number n in the denominator.