AAC #6 Rational exponents and radicals

A rational exponent is an exponent that is a fraction, where the numerator indicates the power and the denominator indicates the root. For example, $$x^{\frac{m}{n}}$$ means the nth root of $$x^m$$ .

To simplify $$\frac{\text{a}}{\text{b}}$$ , express both a and b in terms of their rational exponents, then simplify by subtracting the exponents if they have the same base.

Radicals can be expressed as rational exponents. For example, $$\text{the square root of } x$$ is the same as $$x^{\frac{1}{2}}$$ .

$$\text{a}^{\frac{1}{3} - \frac{1}{4}} = \text{a}^{\frac{4-3}{12}} = \text{a}^{\frac{1}{12}}$$ .

$$\text{a}^{\frac{1}{2}} = \text{sqrt}(16) = 4$$ .

$$\text{a}^{\frac{m}{n}} = \text{the nth root of } \text{a}^m$$ .

$$\text{a}^{\frac{2}{3} + \frac{4}{3}} = \text{a}^{\frac{6}{3}} = \text{a}^{2}$$ .

$$\text{b}^{\frac{2}{3}} = \text{c}^{\frac{3}{2} \times \frac{2}{3}} = \text{c}^{1} = \text{c}$$ .

To simplify an expression with radicals means to rewrite it in a simpler form, often by eliminating the radical or reducing the expression to its lowest terms.

$$\text{sqrt}[3]{x^2 \times x^4} = \text{sqrt}[3]{x^{6}} = x^{2}$$ .