Question

1. reinforce under what conditions will a simplified expression still have a rational exponent? 8. reinforce summarize what a negative exponent does to an expression.

Explanation:

Question 7

A simplified expression will still have a rational exponent when the radicand (the number inside the radical) cannot be written as a perfect power corresponding to the denominator of the rational exponent. For example, if we have \(a^{\frac{m}{n}}\), and \(a\) is not a perfect \(n\) - th power (\(a = b^n\) for some integer \(b\) does not hold), then the expression \(a^{\frac{m}{n}}\) (or \(\sqrt[n]{a^m}\)) cannot be simplified to an integer or a non - rational exponent form.

A negative exponent \(a^{-n}\) (where \(a
eq0\) and \(n\) is a positive real number) means taking the reciprocal of the base raised to the positive of that exponent. Mathematically, \(a^{-n}=\frac{1}{a^{n}}\). For example, \(2^{-3}=\frac{1}{2^{3}}=\frac{1}{8}\).

Answer:

A simplified expression has a rational exponent when the radicand is not a perfect power corresponding to the denominator of the rational exponent.

Question 8