4.0-4.2 Review

The fourth root of a variable raised to a power is expressed as @@ ext{If } k^n, ext{ then } ext{the fourth root is } ext{ } oot{4}{k^n} = k^{rac{n}{4}}@@.

To simplify, use the properties of exponents: @@rac{ oot{3}{x^{18}}}{ oot{2}{x^4}} = rac{x^{rac{18}{3}}}{x^{rac{4}{2}}} = rac{x^6}{x^2} = x^{6-2} = x^4@@.

To rewrite an expression in radical form, identify the exponent and convert it using the radical notation: @@x^{rac{m}{n}} = oot{n}{x^m}@@.

A radical can be expressed in exponential form by converting the root to a fractional exponent: @@ oot{n}{x} = x^{rac{1}{n}}@@.

When dividing exponents with the same base, subtract the exponents: $$\frac{a^m}{a^n} = a^{m-n}$$ .

Use the rule of exponents: $$\frac{x^{\frac{1}{3}}}{x^{-\frac{1}{4}}} = x^{\frac{1}{3} - (-\frac{1}{4})} = x^{\frac{1}{3} + \frac{1}{4}} = x^{\frac{4}{12} + \frac{3}{12}} = x^{\frac{7}{12}}$$ .

A radical expression is an expression that includes a root, such as a square root or cube root, represented as @@ oot{n}{x}@@.

Exponents and roots are inversely related: @@ oot{n}{x^m} = x^{rac{m}{n}}@@.

Convert to radical form: @@x^{rac{1}{2}} = oot{2}{x}@@.

The simplified form is $$x^4$$ .