Problem 125 - College Algebra

Question

What does \(a^{\frac{m}{n}}\) mean?

Step-by-Step Solution

Answer

The expression \(a^{\frac{m}{n}}\) means 'a' raised to the power 'm' and then taking the nth root of the result. Alternatively, it denotes the nth root of 'a' raised to the power 'm'. It combines both the operations of exponentiation and root finding.

Step 1: Recognise the combination of exponent and root

Firstly, realise that \(a^{\frac{m}{n}}\) combines two operations: exponentiation and finding a root. The numerator 'm' is the exponent, whereas the denominator 'n' is the root.

Step 2: Express it in exponential form

Express the number 'a' to the power of 'm' which can be written as \(a^{m}\).

Step 3: Express it in root form

Find the nth root of \(a^m\). This is equivalent to taking the whole result of the expression \(a^m\), and then calculating its nth root. In other words, it can also be expressed as \(\sqrt[n]{a^m}\).