College Algebra Study Guide: Quadratics, Exponents, and Radicals

Quadratics, Exponents, and Radicals

Review of Basic Concepts: Simplifying Radicals and Rationalizing Denominators

This section covers the simplification of radical expressions and the process of rationalizing denominators. Mastery of these skills is essential for manipulating algebraic expressions and solving equations involving roots.

• Radical Expression: An expression that contains a root, such as a square root or cube root.

• Simplifying Radicals: Express the radical in its simplest form by factoring out perfect squares (or cubes, etc.).

• Rationalizing the Denominator: The process of eliminating radicals from the denominator of a fraction.

Example:

Simplify $\sqrt{72}$:

• Factor 72: $72 = 36 \times 2$

• $\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$

Example:

Rationalize $\frac{1}{\sqrt{3}}$:

• Multiply numerator and denominator by $\sqrt{3}$: $\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

Quadratic Equations: Methods of Solution

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. There are several methods to solve them, including factoring, completing the square, and using the quadratic formula.

• Factoring: Express the quadratic as a product of two binomials and set each factor to zero.

• Completing the Square: Rearrange the equation to form a perfect square trinomial.

• Quadratic Formula: For any quadratic $ax^2 + bx + c = 0$, the solutions are given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Discriminant: $b^2 - 4ac$ determines the nature of the roots (real and distinct, real and equal, or complex).

Example:

Solve $x^2 + 6x + 5 = 0$ by factoring:

• $(x + 1)(x + 5) = 0$

• Solutions: $x = -1$, $x = -5$

Exponents: Expressing with Positive Exponents

Expressions with exponents should be simplified to have only positive exponents. This often involves using the properties of exponents and rationalizing denominators.

• Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$

• Product Rule: $a^m \cdot a^n = a^{m+n}$

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

Example:

Express $\frac{(4x^{-2}y^4)^3}{(x^4y^{-2})^2}$ with positive exponents:

• Expand numerator: $(4^3)(x^{-2})^3(y^4)^3 = 64x^{-6}y^{12}$

• Expand denominator: $(x^4)^2(y^{-2})^2 = x^8y^{-4}$

• Combine: $\frac{64x^{-6}y^{12}}{x^8y^{-4}} = 64x^{-14}y^{16} = \frac{64y^{16}}{x^{14}}$

Operations with Radicals

When performing operations with radicals, always express the result in simplest radical form and rationalize denominators if necessary.

• Addition/Subtraction: Combine like radicals (same index and radicand).

• Multiplication: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$

• Division: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ (rationalize denominator if needed)

Example:

Perform $\sqrt{20} + \sqrt{45}$:

• $\sqrt{20} = 2\sqrt{5}$, $\sqrt{45} = 3\sqrt{5}$

• Sum: $2\sqrt{5} + 3\sqrt{5} = 5\sqrt{5}$

Solving Radical Equations

Equations involving radicals can often be solved by isolating the radical and then raising both sides to the appropriate power. Always check for extraneous solutions.

• Isolate the radical on one side of the equation.

• Raise both sides to the power that eliminates the radical.

• Solve the resulting equation.

• Check all solutions in the original equation.

Example:

Solve $\sqrt{5x + 9} + 1 = x$:

• Isolate: $\sqrt{5x + 9} = x - 1$

• Square both sides: $5x + 9 = (x - 1)^2$

• Expand: $5x + 9 = x^2 - 2x + 1$

• Rearrange: $x^2 - 7x - 8 = 0$

• Factor: $(x - 8)(x + 1) = 0$

• Solutions: $x = 8$, $x = -1$ (check both in original equation)

Summary Table: Properties of Exponents and Radicals

Property	Expression	Example
Product of Radicals	$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$	$\sqrt{2} \cdot \sqrt{8} = \sqrt{16} = 4$
Quotient of Radicals	$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$	$\frac{\sqrt{18}}{\sqrt{2}} = \sqrt{9} = 3$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$x^{-3} = \frac{1}{x^3}$
Rational Exponent	$a^{1/n} = \sqrt[n]{a}$	$8^{1/3} = 2$

Additional info:

• Always check for extraneous solutions when solving radical equations, as squaring both sides can introduce invalid solutions.

• When rationalizing denominators, multiply numerator and denominator by the radical needed to clear the denominator.