Roots, Rational Exponents, and Equations: College Algebra Study Notes

Roots and Rational Exponents

nth Roots and Radicals

The principal nth root of a number a is written as $\sqrt[n]{a}$ and is defined as the number b such that $b^n = a$. The value of n determines the type of root:

• Square root ($n=2$): $\sqrt{a}$

• Cube root ($n=3$): $\sqrt[3]{a}$

• For $n$ even, $a \geq 0$ and $b \geq 0$

• For $n$ odd, $a$ can be any real number

Examples:

• $\sqrt[3]{8} = 2$ because $2^3 = 8$

• $\sqrt[4]{16} = 2$ because $2^4 = 16$

• $\sqrt[5]{-32} = -2$ because $(-2)^5 = -32$2

Properties:

• If $n$ is odd: $\sqrt[n]{a} = a$

• If $n$ is even: $\sqrt[n]{a} = |a|$

Simplifying Radicals

Radicals can often be simplified using properties of exponents and factorization.

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

• Example: $\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}$

• Example: $\sqrt{54} = \sqrt{27 \cdot 2} = \sqrt{27} \cdot \sqrt{2} = 3\sqrt{6}$

• Example: $\sqrt{12x^3y^2} = \sqrt{4 \cdot 3 \cdot x^2 \cdot x \cdot y^2} = 2xy\sqrt{3x}$

Rationalizing Denominators

To rationalize the denominator means to eliminate any radicals from the denominator of a fraction.

• Example 1: $\frac{5}{\sqrt{3}} = \frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$

• Example 2: $\frac{3}{\sqrt{2}} = \frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2}$

• Example 3 (with conjugates): $\frac{3}{\sqrt{7} - \sqrt{2}}$

  • Multiply numerator and denominator by the conjugate $\sqrt{7} + \sqrt{2}$:

  • $\frac{3}{\sqrt{7} - \sqrt{2}} \cdot \frac{\sqrt{7} + \sqrt{2}}{\sqrt{7} + \sqrt{2}} = \frac{3(\sqrt{7} + \sqrt{2})}{(\sqrt{7} - \sqrt{2})(\sqrt{7} + \sqrt{2})}$

  • Denominator: $(\sqrt{7})^2 - (\sqrt{2})^2 = 7 - 2 = 5$

  • Final answer: $\frac{3(\sqrt{7} + \sqrt{2})}{5}$

Rational Exponents

Rational exponents provide an alternative way to write roots and powers.

• For any real number a and integer $n \geq 2$:

• $a^{1/n} = \sqrt[n]{a}$

• For a rational exponent $m/n$ in lowest terms:

$a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$

• Examples:

• $\sqrt[2]{8^2} = 2^2 = 4$

• $(\sqrt[2]{16})^3 = 2^3 = 8$

• $25^{1/2} = 5$

• The same exponent laws apply to rational exponents as to integer exponents.

Summary on Polynomials, Factoring, and Rational Expressions

Polynomials

A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

• Examples: $x^2 + 3x + 2$, $4y^3 - y + 7$

• Special products: $(a+b)^2 = a^2 + 2ab + b^2$

Factoring

Factoring is the process of writing a polynomial as a product of its factors.

• Methods include:

• Factoring by grouping

• Factoring trinomials

• Factoring special forms (difference of squares, perfect square trinomials, sum/difference of cubes)

Rational Expressions

A rational expression is a ratio of two polynomials. Simplification involves factoring and reducing common factors.

• Example: $\frac{x^2 - 9}{x^2 - x - 6} = \frac{(x+3)(x-3)}{(x-3)(x+2)} = \frac{x+3}{x+2}$ (for $x \neq 3$)

Equations and Inequalities

Linear Equations

A linear equation is an equation of the form $ax + b = c$ where $a \neq 0$.

• Solving steps:

• Isolate the variable on one side of the equation.

• Perform inverse operations (addition, subtraction, multiplication, division).

Examples:

• $x + 2 = 8$ Subtract 2: $x = 6$

• $\frac{x}{2} = 5$ Multiply both sides by 2: $x = 10$

• $\frac{3x}{x-1} = 6$ Multiply both sides by $(x-1)$: $3x = 6(x-1)$ Expand: $3x = 6x - 6$ Subtract $6x$: $-3x = -6$ Divide by $-3$: $x = 2$

Steps for Solving Applied Linear Equations

1. Read the problem carefully.

2. Assign a variable to represent the unknown.

3. List all known facts.

4. Write and solve the equation.

5. Check the answer in the context of the problem.

Example 1: $18,000 is invested in stocks and bonds. The amount in bonds is half that in stocks. How much is invested in each?

• Let $x$ = amount in stocks, amount in bonds = $\frac{1}{2}x$

• Total: $x + \frac{1}{2}x = 18000$

• $\frac{3}{2}x = 18000 \implies x = 12000$ (stocks), $6000$ (bonds)

Example 2: At what temperature are Celsius and Fahrenheit readings the same?

• Formula: $F = \frac{9}{5}C + 32$

• Set $F = C = x$

• $x = \frac{9}{5}x + 32$

• $5x = 9x + 160$

• $-4x = 160 \implies x = -40$

• So, $-40^\circ$ is the same in both scales.

Quadratic Equations

A quadratic equation is of the form $Ax^2 + Bx + C = 0$ where $A \neq 0$.

• Methods of solution:

• Factoring

• Quadratic Formula

• Completing the Square

Solving by Factoring

• Example: $x^2 + 5x + 6 = 0$

• Factor: $(x+2)(x+3) = 0$

• Solutions: $x = -2$, $x = -3$

Quadratic Formula

For $Ax^2 + Bx + C = 0$, the solutions are:

$x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$

• Example: $x^2 + 5x + 6 = 0$ $A=1, B=5, C=6$ $x = \frac{-5 \pm \sqrt{25 - 24}}{2} = \frac{-5 \pm 1}{2}$ So $x = -2$ or $x = -3$

Completing the Square

• Rewrite $x^2 + 5x + 6 = 0$ as $x^2 + 5x = -6$

• Add $(\frac{5}{2})^2 = 6.25$ to both sides: $x^2 + 5x + 6.25 = 0.25$

• $(x + 2.5)^2 = 0.25$

• $x + 2.5 = \pm 0.5$

• $x = -2$ or $x = -3$

• Additional info: The notes use a different example, but the process is the same.

Applied Quadratic Problem Example

A box has height 9 cm, length $(x-12)$ cm, width $(x-18)$ cm, and volume 144 cm3. Find $x$.

• Volume: $9(x-12)(x-18) = 144$

• $(x-12)(x-18) = 16$

• $x^2 - 30x + 216 = 16$

• $x^2 - 30x + 200 = 0$

• Factor: $(x-10)(x-20) = 0$

• Possible $x$: 10 or 20

• Check: $x=10$ gives negative length, so $x=20$ is valid

Radical Equations, Quadratic in Form, and Factorable Equations

Some equations involve radicals or are quadratic in form (i.e., can be rewritten as a quadratic in another variable).

• Radical Equations: Equations involving roots, e.g., $\sqrt{x} = 3$

• Quadratic in Form: Equations like $y^2 + 3y - 10 = 0$ (let $u = y$)

• Example: $u^2 + 3u - 10 = 0$ Factor: $(u+5)(u-2) = 0$ So $u = -5$ or $u = 2$ Substitute back for $y$

Factorable Equations

Equations that can be solved by factoring and setting each factor to zero.

• Example: $x^3 - 8x = 0$ Factor: $x(x^2 - 8) = 0$ So $x = 0$, $x = 2\sqrt{2}$, $x = -2\sqrt{2}$

Summary Table: Methods for Solving Equations

Equation Type	Method(s)	Example
Linear	Isolate variable, inverse operations	$2x + 3 = 7$
Quadratic	Factoring, Quadratic Formula, Completing the Square	$x^2 + 5x + 6 = 0$
Radical	Isolate radical, square both sides	$\sqrt{x} = 3$
Quadratic in Form	Substitution, then solve as quadratic	$y^2 + 3y - 10 = 0$
Factorable	Factor and set each factor to zero	$x^3 - 8x = 0$

Key Takeaways

• Understand and apply properties of roots and rational exponents.

• Simplify and manipulate radical expressions.

• Rationalize denominators using appropriate techniques.

• Solve linear, quadratic, radical, and quadratic-in-form equations using systematic methods.

• Check solutions in the context of the original problem, especially for extraneous solutions in radical equations.