Fractional Exponents and Solving Linear Inequalities

Vocabulary practice flashcards covering the definitions, properties, and procedural rules of fractional exponents, radical simplification, rationalization, and graphing linear inequalities.

Fractional exponent definition ($a^{\frac{1}{n}}$)

The formal mathematical definition stating that $a^{\frac{1}{n}} = \sqrt[n]{a}$, where $n \ne 0$ and $n$ is an integer.

General fractional exponent rule ($a^{\frac{m}{n}}$)

The algebraic property establishing that $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$.

Power of a power rule for fractional exponents

The property showing that $(a^{\frac{1}{n}})^n = a^{\frac{n}{n}} = a^1 = a$, and $(a^{\frac{1}{n}})^m = a^{\frac{m}{n}}$.

Product rule for roots

The rule stating that $\sqrt[n]{a}\sqrt[n]{b} = \sqrt[n]{ab}$, which follows from the exponent property $a^x b^x = (ab)^x$.

Quotient rule for roots

The rule stating that $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$, which follows from the exponent property $\frac{a^x}{b^x} = \left(\frac{a}{b}\right)^x$.

Principal root convention

The definition rule stating that if two values satisfy an even-root equation (such as $x^4 = 16$ having solutions $2$ and $-2$), the positive value is used, making $\sqrt[4]{16} = 2$.

Even root of a negative radicand

An expression such as $(-36)^{\frac{1}{4}} = \sqrt[4]{-36}$ where no real number solution exists to $x^4 = -36$, meaning it is not a real number.

Negative fractional exponent rule

The rule for evaluating a negative exponent by placing the base and positive exponent in the denominator, expressed as $a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}}$.

Rationalizing a denominator

The process of eliminating all radical signs from the denominator of a fraction.

Multiplying radicals with different indices

The method of converting radicals to fractional exponents, finding a least common denominator (LCD) to add or rewrite the fractional exponents, and re-expressing the result under a common root index.

Simplifying radicals by factoring the radicand

A simplification method where the number under the radical is factored into exponential form so the fractional exponent can be reduced (e.g., $\sqrt[6]{8} = 8^{\frac{1}{6}} = (2^3)^{\frac{1}{6}} = 2^{\frac{3}{6}} = 2^{\frac{1}{2}} = \sqrt{2}$).

Linear programming

A branch of mathematics concerned with solving practical problems involving linear inequalities, used in fields such as business, economics, biology, and astronomy.

Half-plane

The geometric representation of the solution to a linear inequality on a coordinate plane, consisting of an infinite set of points lying on one side of a boundary line.

Dotted boundary line

A dashed line used when graphing strict inequalities ($<$ and $>$) to indicate that points on the boundary line are not included in the solution set.

Solid boundary line

A continuous line used when graphing non-strict inequalities ($\le$ and $\ge$) to indicate that points on the boundary line are included in the solution set.

Steps for graphing a linear inequality

A three-step procedure: 1. Solve the inequality for $y$; 2. Graph the boundary line (solid for $\le$/$\ge$ or dotted for $<$/$>$); 3. Shade the appropriate half-plane (above the line for $>$/$\ge$ or below the line for $<$/$\le$).