Fundamental Properties and Techniques in Precalculus Algebra

Properties of Exponents

Introduction to Exponents

An exponent is a shorthand notation for repeated multiplication of a number by itself. For example, $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ can be written as $2^5$. In this expression, 2 is the base and 5 is the exponent.

• Exponential Expression: An expression of the form $x^n$, where $x$ is a real number and $n$ is a positive integer.

• Expanded Form: $x^n = \underbrace{x \cdot x \cdot x \cdots x}_{n \text{ factors of } x}$

Product Rule for Exponents

If $m$ and $n$ are positive integers and $a$ is a real number, then:

• $a^m \cdot a^n = a^{m+n}$

• Example: $2^3 \cdot 2^4 = 2^{3+4} = 2^7$

Power Rule for Exponents

If $m$ and $n$ are positive integers and $a$ is a real number, then:

• $(a^m)^n = a^{mn}$

• Example: $(3^2)^4 = 3^{2 \times 4} = 3^8$

Power of a Product Rule

If $n$ is a positive integer and $a$ and $b$ are real numbers, then:

• $(ab)^n = a^n b^n$

• Example: $(2 \cdot 5)^3 = 2^3 \cdot 5^3 = 8 \cdot 125 = 1000$

Power of a Quotient Rule

If $n$ is a positive integer, $a$ and $b$ are real numbers, and $b \neq 0$, then:

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

• Example: $\left(\frac{3}{2}\right)^4 = \frac{3^4}{2^4} = \frac{81}{16}$

Quotient Rule for Exponents

If $m$ and $n$ are positive integers, $a$ is a real number, and $a \neq 0$, then:

• $\frac{a^m}{a^n} = a^{m-n}$

• Example: $\frac{5^7}{5^3} = 5^{7-3} = 5^4$

Zero Exponent Rule

If $b$ is a real number such that $b \neq 0$, then:

• $b^0 = 1$

• Example: $7^0 = 1$

Linear Equations and Their Properties

Definition of a Linear Equation

A linear equation in one variable is an equation that can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$.

• Examples of Linear Equations:

  • $\frac{7}{2} - 4x = x$

  • $\sqrt{5x - 3} = 8x - 1$

  • $(0.7)x + 1 = 6$

  • $2^1 - 0.4x = 0.3 - 5x$

• Examples of Non-linear Equations:

  • $\frac{3}{x} = 5 - 9$

  • $\sqrt{x^2 + 1} = 7$

  • $x^2 + x + 6 = (0.7)x^2 + x = 1$

  • $0.3 - 0.2x^2 = 2$

Simplifying Algebraic Expressions

Distributive Property and Like Terms

An algebraic expression containing the sum or difference of like terms can be simplified by applying the distributive property. This is called combining like terms.

• Distributive Property: $a(b + c) = ab + ac$

• Example: $3x + 2x = (3 + 2)x = 5x$

• When simplifying expressions containing parentheses, use the distributive property to remove parentheses and then combine like terms.

Properties of Equality

Addition and Multiplication Properties

• Addition Property of Equality: If $a$, $b$, and $c$ are real numbers and $a = b$, then $a + c = b + c$

• Multiplication Property of Equality: If $a$, $b$, and $c$ are real numbers, $c \neq 0$, and $a = b$, then $ac = bc$

• Multiplying or dividing both sides of an equation by the same nonzero number preserves equality.

Solving Linear Equations with Integer Coefficients

Isolating the Variable

To solve for $x$ in a linear equation, manipulate the equation so that $x$ is isolated on one side. Substitute the solution back into the original equation to verify correctness.

• Goal: Isolate the variable $x$.

• Method: Use addition, subtraction, multiplication, or division properties of equality.

Finding a Least Common Denominator (LCD)

Definition and Application

Given a set of fractions, the least common denominator is the smallest number that is divisible by each denominator.

• Example: For $\frac{1}{4}$ and $\frac{1}{6}$, the LCD is 12.

• Use the LCD to combine fractions or solve equations involving fractions.

Solving Linear Equations Involving Decimals

Eliminating Decimals

To solve linear equations with decimals, multiply both sides by the smallest power of 10 that eliminates all decimals. This ensures the new equation contains only integers.

• Strategy: Identify the greatest number of decimal places in the equation, then multiply both sides by $10^n$, where $n$ is the number of decimal places.

• Example: For $0.3x + 0.25 = 0.05x$, multiply both sides by 100 to get $30x + 25 = 5x$.