Radicals and Rational Exponents Notes

Perfect Squares, Cubes, and Fourth Powers

• Perfect Squares: Numbers that result from squaring an integer. Examples include $2^2 = 4$, $3^3 = 9$, and $4^4 = 16$.

• Perfect Cubes: Numbers that result from cubing an integer. Examples include $2^3 = 8$, $3^3 = 27$.

• Fourth Power: Numbers raised to the fourth power (e.g., $2^4$).

Radicals and Radicands

• Index: The small number indicating the root to be taken (understood to be 2 for square root).

• Radicand: The number or expression under the radical symbol.

Radical Rules

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

• Quotient Rule: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

• Important Note: $\sqrt{a \pm b} \neq \sqrt{a} \pm \sqrt{b}$. Example: $\sqrt{25 + 4} \neq \sqrt{25} + \sqrt{4}$

• Nested Radical: $\sqrt[a]{\sqrt[b]{x}} = \sqrt[a \cdot b]{x}$

• Square Root of $x^2$: $\sqrt{x^2} = |x|$

Simplifying Radicals

• Simplifying Strategy: Break down the radicand into factors, identifying perfect squares, cubes, etc.

  • Example: $\sqrt{72x^3} = \sqrt{36 \cdot 2 \cdot x^2 \cdot x} = \sqrt{36} \cdot \sqrt{x^2} \cdot \sqrt{2x} = 6|x|\sqrt{2x}$

• Absolute Value: Remember to use absolute value when simplifying radicals with even indices and variable radicands.

  • Example: $\sqrt{9x^2} = 3|x|$

Combining Radicals

• Like Radicals: Radicals with the same index and radicand can be added or subtracted.

  • Example: $\sqrt{20} + 6\sqrt{5} = 2\sqrt{5} + 6\sqrt{5} = 8\sqrt{5}$

• Simplifying Before Combining: Simplify each radical before attempting to combine like terms.

  • Example: $4\sqrt{12} - 2\sqrt{75} = 4\sqrt{4 \cdot 3} - 2\sqrt{25 \cdot 3} = 8\sqrt{3} - 10\sqrt{3} = -2\sqrt{3}$

Cube Roots

• Simplifying Cube Roots: Look for perfect cube factors within the radicand.

  • Example: $\sqrt[3]{54} = \sqrt[3]{27 \cdot 2} = 3\sqrt[3]{2}$

• Combining Cube Roots: Similar to square roots, combine like cube roots.

  • Example: $\sqrt[3]{24} + \sqrt[3]{81} = 2\sqrt[3]{3} + 3\sqrt[3]{3} = 5\sqrt[3]{3}$

Rationalizing Denominators

• Goal: Eliminate radicals from the denominator.

• Method: Multiply the numerator and denominator by a suitable expression.

  • Single Radical Term: Multiply by the radical itself.

    • Example: $\frac{3}{\sqrt{10}} = \frac{3\sqrt{10}}{\sqrt{10}\sqrt{10}} = \frac{3\sqrt{10}}{10}$

  • Binomial Denominator: Multiply by the conjugate.

    • Example: $\frac{3}{\sqrt{3} + 1} = \frac{3(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} = \frac{3\sqrt{3} - 3}{3 - 1} = \frac{3\sqrt{3} - 3}{2}$

Rational Exponents

• Rule: $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

• Example: $16^{\frac{1}{4}} = \sqrt[4]{16} = 2$

Solving Absolute Value Equations

• Absolute Value: Represents distance from zero.

• Two Cases: For $|x| = a$, solve for both $x = a$ and $x = -a$.

• Isolate: Before splitting into two equations, isolate the absolute value expression.

  • Example:
    $7|3x| + 3 = 16$
    $7|3x| = 14$
    $|3x| = 2$
    $3x = 2$ or $3x = -2$
    $x = \frac{2}{3}$ or $x = -\frac{2}{3}$

• Checking Answers: Always check solutions in the original equation.

Absolute Value Inequalities

• Less Than: Represents an intersection (AND).

• Greater Than: Represents a union (OR).

• Less Than (AND): For $|x| < a$, solve $-a < x < a$.

• Greater Than (OR): For $|x| > a$, solve $x < -a$ or $x > a$.

• Example: Solve $|3x + 5| > 7$
  $3x + 5 < -7$ or $3x + 5 > 7$
  $3x < -12$ or $3x > 2$
  $x < -4$ or $x > \frac{2}{3}$

Functions and Graphs

• Relation: A set of ordered pairs.

• Function: A relation where each element in the domain is paired with exactly one element in the range (no repeated x-values).

• Domain: Input values (x-values).

• Range: Output values (y-values).

Representing Functions

• Table: Listing of ordered pairs.

• Graph: Visual representation of ordered pairs.

• Mapping Diagram: Illustrates the mapping from domain to range.

Vertical Line Test

• Purpose: Determines if a graph represents a function. A graph is a function if no vertical line intersects it more than once.

Function Notation

• Notation: $f(x)$

• f: Name of the function.

• x: Input variable.

• f(x): Expression representing the output.

Evaluating Functions

• Substitution: Replace the input variable with the given value or expression.

  • Example: If $f(x) = 3x + 7$, then $f(4) = 3(4) + 7 = 19$

Reading Graphs

• x-intercepts: Points where the graph crosses the x-axis.

• y-intercepts: Points where the graph crosses the y-axis.

• Domain: Set of all x-values for which the function is defined.

• Range: Set of all y-values that the function takes.

• Evaluating from Graph: Find the y-value for a given x-value on the graph.

Combinations of Functions

• Addition: $(f + g)(x) = f(x) + g(x)$

• Subtraction: $(f - g)(x) = f(x) - g(x)$

• Multiplication: $(f \cdot g)(x) = f(x) \cdot g(x)$

• Division: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, where $g(x) \neq 0$

Composition of Functions

• Definition: $(f \circ g)(x) = f(g(x))$

• Evaluation: Evaluate the inner function first, then use the result as the input for the outer function.

Domain Restrictions

• Denominators: Exclude x-values that make the denominator zero.

• Square Roots: Exclude x-values that result in a negative radicand.

Examples of Combined Function Operations

• Given $f(x) = 3x - 4$ and $g(x) = x + 2$

  • $(f + g)(x) = 4x - 2$

  • $(f - g)(x) = 2x - 6$

  • $(f \cdot g)(x) = 3x^2 + 2x - 8$

  • $\left(\frac{f}{g}\right)(x) = \frac{3x - 4}{x + 2}$, $x \neq -2$

  • $(f \circ g)(x) = 3(x+2) - 4 = 3x + 2$

  • $(g \circ f)(x) = (3x - 4) + 2 = 3x - 2$