Problem 125 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The expression \(a^{\frac{m}{n}}\) means that \(a\) is raised to the power of \(m\), and then the \(n\)th root of that result is taken, which can be expressed like \(\sqrt[n]{a^m}\) or \(\sqrt[n]{a}^m\).

Step 1: Identify the base of the exponent

The base of the exponent is \(a\), which can be any non-zero number.

Step 2: Understand the numerator of the exponent

The numerator \(m\) of the fractional exponent is the power that the base is raised to. This simply means that \(a\) is multiplied by itself \(m\) times.

Step 3: Understand the denominator of the exponent

The denominator \(n\) of the fractional exponent represents the type of root of the base. That is, if \(n\) is 2, it represents a square root, if \(n\) is 3, it represents a cube root, and so on.

Step 4: Tie everything together

Thus, corresponds to the \(n\)th root of \(a\) raised to the \(m\)th power, and it can be expressed as \(\sqrt[n]{a^m}\) or \(\sqrt[n]{a}^m\).