Math 1013 College Algebra Review

Negative Exponent Rule

The algebraic property stating that $a^{-n} = \frac{1}{a^n}$ , used to simplify expressions such as $\frac{2x^{-2}y^3}{10xy^{-2}} = \frac{y^5}{5x^3}$ and $(3m)^{-3/2} = \frac{1}{\sqrt{(3m)^3}}$.

Rational Exponent

An exponent of the form $\frac{m}{n}$, defined by $a^{m/n} = (\sqrt[n]{a})^m$, applied in simplifying terms such as $-27^{2/3} = -(\sqrt[3]{27})^2 = -(3)^2 = -9$.

Imaginary Unit ($i$)

A mathematical unit defined by $i = \sqrt{-1}$ and $i^2 = -1$, used to evaluate radicals with negative radicands like $\sqrt{-20a^5} = 2a^2i\sqrt{5a}$ and complex expressions like $(\sqrt{8} - i\sqrt{2})^2$.

Extraneous Solution

A false solution that emerges from the algebraic solving process—such as squaring both sides of a radical equation like $\sqrt{x-1} + 7 = x$—which does not satisfy the original equation.

Conditional Equation

An algebraic equation that is true for only a specific, finite set of values, such as $$4(x+3