College Algebra Review: Numbers, Operations, Exponents, Radicals, and Polynomials

Important Sets of Numbers

Classification of Numbers

In mathematics, numbers are classified into several important sets based on their properties and uses. Understanding these sets is foundational for algebraic reasoning.

• Natural Numbers: The counting numbers, given by {1, 2, 3, ...}.

• Whole Numbers: The counting numbers including zero, {0, 1, 2, 3, ...}.

• Integers: The set of whole numbers and their negatives, {..., -2, -1, 0, 1, 2, ...}.

• Rational Numbers: Numbers that can be written as a fraction with an integer numerator and a nonzero integer denominator (e.g., $\frac{3}{2}$, $-\frac{5}{7}$, $0.25$).

• Irrational Numbers: Numbers that cannot be written as a fraction of integers. Their decimal representations are non-terminating and non-repeating (e.g., $\sqrt{2}$, $\pi$).

• Real Numbers: The set of all rational and irrational numbers.

Example: $\frac{3}{2}$ is rational, $\sqrt{2}$ is irrational, $-5$ is an integer, $0$ is a whole number.

Order of Operations (PEMDAS)

Evaluating Expressions

To correctly evaluate mathematical expressions, follow the order of operations, often remembered by the acronym PEMDAS:

1. Parentheses: Perform operations inside parentheses or brackets first.

2. Exponents: Evaluate powers and roots.

3. Multiplication and Division: From left to right.

4. Addition and Subtraction: From left to right.

Example: $(-2) + 3 - 4(5 - 7) = -2 + 3 - 4(-2) = -2 + 3 + 8 = 9$

Properties of Exponents

Exponent Rules

Exponents are used to represent repeated multiplication. The following properties are essential for simplifying expressions:

• $b^m \cdot b^n = b^{m+n}$

• $\frac{b^m}{b^n} = b^{m-n}$

• $(b^m)^n = b^{mn}$

• $(ab)^n = a^n b^n$

• $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

• $b^0 = 1$ (for $b \neq 0$)

Example: $4^5 = 1024$, $5^3 = 125$, $2^6 = 64$

Properties of Square Roots

Square Root Rules

The square root of a nonnegative number $a$ is the nonnegative number $b$ such that $b^2 = a$. The following properties are useful:

• $\sqrt{a^2} = |a|$

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

• $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ (for $b \neq 0$)

Note: The square root of a negative real number is not a real number.

Example: $\sqrt{8} = 2\sqrt{2}$, $\sqrt{-49}$ is not a real number, $\sqrt{49} = 7$

Operations with Fractions

Adding, Subtracting, Multiplying, and Dividing Fractions

Fractions are manipulated using the following rules:

• Add/Subtract: Find a common denominator, then add or subtract numerators.

• Multiply: Multiply numerators and denominators directly.

• Divide: Multiply by the reciprocal of the divisor.

Examples:

• $\frac{1}{5} + \frac{4}{5} = \frac{5}{5} = 1$

• $\frac{2}{9} - \frac{4}{7} = \frac{14 - 36}{63} = -\frac{22}{63}$

• $\frac{1}{4} \times \frac{3}{5} = \frac{3}{20}$

• $\frac{4}{5} \div \frac{3}{5} = \frac{4}{5} \times \frac{5}{3} = \frac{4}{3}$

• $2 + \frac{1}{3} = \frac{7}{3}$

Simplifying Radicals

Radical Expressions

To simplify a radical such as $\sqrt{a}$, factor out the largest perfect square from $a$ and take the square root of that factor.

• $\sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7}$

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

• $\sqrt{147} = \sqrt{49 \times 3} = 7\sqrt{3}$

Roots and Fractional Exponents

Evaluating Roots and Using Fractional Exponents

The $n$th root of $a$ is written $\sqrt[n]{a}$ and can be expressed as a fractional exponent: $a^{1/n}$. For any real number $a$ and integers $m, n$:

• $a^{m/n} = (a^{1/n})^m = (\sqrt[n]{a})^m$

• If $n$ is even, $a$ must be nonnegative for $\sqrt[n]{a}$ to be real.

• If $n$ is odd, $\sqrt[n]{a}$ is defined for all real $a$.

Examples:

• $\sqrt[5]{32} = 2$

• $\sqrt[3]{-27} = -3$

• $(625^4)^{1/4} = 625$

• $(-4)^5 = -1024$

Evaluating Algebraic Expressions

Substitution and Simplification

To evaluate an algebraic expression, substitute the given value for the variable and follow the order of operations.

• Always use parentheses when substituting negative numbers.

Examples:

• For $x = -3$, $x^2 - 5x + 2 = (-3)^2 - 5(-3) + 2 = 9 + 15 + 2 = 26$

• For $x = -3$, $\frac{8 - 4(x)}{2x^2 + 1} = \frac{8 - 4(-3)}{2(-3)^2 + 1} = \frac{8 + 12}{18 + 1} = \frac{20}{19}$

• $|2x - 5|$ for $x = -3$: $|2(-3) - 5| = |-6 - 5| = |-11| = 11$

Polynomials

Definition and Simplification

A polynomial is an expression of the form $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$, where the $a_i$ are real numbers and $n$ is a nonnegative integer. The $a_i$ are called coefficients, and each $a_ix^i$ is a term.

• Multiplying Polynomials: Use the distributive property to expand products.

• Simplifying: Combine like terms and apply exponent rules.

Examples:

• $(3x^2)(2x^3) = 6x^{2+3} = 6x^5$

• $\frac{12x^7}{4x^2} = 3x^{7-2} = 3x^5$

• $(4x^5)^3 = 64x^{15}$

• $\left(\frac{3}{5}x^2\right)^2 = \frac{9}{25}x^4$

• $(2x - 3x + 6) + (4x^2 - 8x + 7) = 2x - 3x + 6 + 4x^2 - 8x + 7 = 4x^2 - 9x + 13$

• $(x^3 + 4x) = x^3 + 4x$

• $5x(x^2 - 3x + 7) = 5x^3 - 15x^2 + 35x$

• $(3x + 5)(4x - 7) = 12x^2 - 21x + 20x - 35 = 12x^2 - x - 35$

Summary Table: Properties of Exponents and Roots

Property	Formula	Example
Product of Powers	$a^m \cdot a^n = a^{m+n}$	$2^3 \cdot 2^4 = 2^{3+4} = 128$
Quotient of Powers	$\frac{a^m}{a^n} = a^{m-n}$	$5^6 / 5^2 = 5^{6-2} = 625$
Power of a Power	$(a^m)^n = a^{mn}$	$(3^2)^3 = 3^{2\times3} = 729$
Power of a Product	$(ab)^n = a^n b^n$	$(2\cdot3)^2 = 2^2 \cdot 3^2 = 36$
Power of a Quotient	$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$	$\left(\frac{2}{5}\right)^3 = \frac{8}{125}$
Zero Exponent	$a^0 = 1$	$7^0 = 1$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$2^{-3} = \frac{1}{8}$
Fractional Exponent	$a^{m/n} = (\sqrt[n]{a})^m$	$8^{2/3} = (\sqrt[3]{8})^2 = 4$

Key Terms

• Variable: A symbol (often a letter) that represents a number whose value can change.

• Algebraic Expression: A combination of numbers, variables, and operations.

• Coefficient: The numerical factor in a term of a polynomial.

• Term: A single number or variable, or numbers and variables multiplied together, separated by + or - signs in an expression.