Roots, Powers, and Rational Exponents

Vocabulary and key concepts covering number classifications, exponent laws, and evaluations of positive rational exponents.

Product of Powers Law

An exponent law stating that $a^m \cdot a^n = a^{m+n}$.

Quotient of Powers Law

An exponent law stating that $a^m \div a^n = a^{m-n}$.

Power of a Power Law

An exponent law stating that $(a^m)^n = a^{m \cdot n}$.

Power of a Quotient Law

An exponent law stating that $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.

Power with a Zero Exponent Law

An exponent law stating that $a^0 = 1$.

Power with a Negative Exponent Law

An exponent law stating that $a^{-n} = \frac{1}{a^n} = \left(\frac{1}{a}\right)^n$.

Powers with Positive Rational Exponents

Represented as $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ or $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$, where $a$ is a rational number, and $m$ and $n$ are natural numbers.

Index of a Radical

The denominator of the rational exponent, represented by $n$ in $\sqrt[n]{a}$ or $a^{\frac{m}{n}}$.

Radicand

The number or expression under the radical sign, represented by $a$ in $\sqrt[n]{a}$.

Classification of $\sqrt{0.0016}$

A rational number because it evaluates to $0.04$.

Classification of $\sqrt{8}$

An irrational number.

Writing Mixed Radicals as Powers with Rational Exponents ($4\sqrt{3}$)

$$4\sqrt{3} = \sqrt{3 \cdot 4 \cdot 4} = \sqrt{48} = 48^{\frac{1}{2}}$$

Evaluation of $100^{1.5}$

$$100^{1.5} = 100^{\frac{3}{2}} = (\sqrt{100})^3 = (10)^3 = 1000$$

Evaluation of $(-8)^{\frac{4}{3}}$

$$(-8)^{\frac{4}{3}} = (\sqrt[3]{-8})^4 = (-2)^4 = 16$$

Evaluation of $\left(\frac{16}{54}\right)^{\frac{2}{3}}$

Simplify the base first to $\left(\frac{8}{27}\right)^{\frac{2}{3}} = \left(\sqrt[3]{\frac{8}{27}}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9}$

Evaluation of $-32^{1.2}$

$$-32^{1.2} = -32^{\frac{6}{5}} = -(\sqrt[5]{32})^6 = -(2)^6 = -64$$