Introduction to Algebraic Expressions: Exponents, Expressions, and Simplification

Evaluating Exponents

Intro to Exponents

Exponents are a way to represent repeated multiplication of the same number. The number being multiplied is called the base, and the number of times it is multiplied is called the exponent.

• Base: The number being multiplied.

• Exponent: Indicates how many times the base is multiplied by itself.

For example, $a^n$ means that the base $a$ is multiplied by itself $n$ times.

Expression	Expanded Form	Value
$2^3$	$2 imes 2 imes 2$	$8$
$5^2$	$5 imes 5$	$25$
$10^1$	$10$	$10$

Note: A number with no exponent shown is understood to have an exponent of 1, e.g., $a = a^1$.

Evaluating Expressions

Algebraic Expressions

An algebraic expression consists of numbers, variables, and operations. Key terms include:

• Variable: A letter that represents any number/value.

• Coefficient: The number multiplied by a variable.

• Constant: A number without a variable.

Example: In $2x + 5$, $x$ is the variable, $2$is the coefficient, and $5$ is the constant.

Evaluating Algebraic Expressions

To evaluate an algebraic expression, substitute the given value(s) for the variable(s) and perform the operations.

• Example: If $x = 3$, evaluate $2x + 5$: $2(3) + 5 = 6 + 5 = 11$

• Example: If $a = 4$, $b = 2$, evaluate $a^2 + 3b$: $4^2 + 3(2) = 16 + 6 = 22$

Translating Phrases to Expressions

Translating Word Problems

Algebraic expressions can be formed by translating verbal phrases using mathematical operations. Common keywords help identify the operation:

Operation/Symbol	Common Keywords	Example
Variable	A number, a quantity, an unknown value	Five more than a number
Addition (+)	Sum, more than, plus	Five more than a number: $x + 5$
Subtraction (−)	Difference, less than, minus	A number decreased by seven: $x - 7$
Multiplication (×)	Product, times, multiplied by	The product of an unknown and 6: $6x$
Division (÷)	Quotient, divided by, per	Seven divided by a number: $7/x$

• Example: The quotient of a number and three: $x/3$

• Example: Negative three times the value of $x$ increased by 5: $-3(x + 5)$

Simplifying Expressions

Like Terms

Algebraic expressions often contain like terms, which are terms that have the same variable(s) raised to the same power(s). Only like terms can be combined by addition or subtraction.

Like Terms	Not Like Terms
$2x^2$, $5x^2$	$2x^2$, $5x$
$3y$, $-7y$	$3y$, $-7z$
$4$,$-2$	$4$,$-2x$

• Example: Combine $4x^2 + 7x^2$: $4x^2 + 7x^2 = 11x^2$

• Example: Combine $3a + 10a$: $3a + 10a = 13a$

Simplifying Algebraic Expressions

To simplify an algebraic expression:

1. Distribute multiplication if needed.

2. Identify like terms (same variable and exponent).

3. Group like terms by addition or subtraction.

4. Combine like terms.

• Example: Simplify $2a + 2b + 5a$: $2a + 5a + 2b = 7a + 2b$

• Example: Simplify $3x^2 + 5x^2$: $3x^2 + 5x^2 = 8x^2$

Practice Problems

• Simplify $7x - 2x + 10$: $7x - 2x = 5x$, so $5x + 10$

• Simplify $3a^2 + 5a^2$: $3a^2 + 5a^2 = 8a^2$

• Simplify $8.6 + 2.5x - 1.35 + 1.55$: Combine constants: $8.6 - 1.35 + 1.55 = 8.8$, so $8.8 + 2.5x$

Summary Table: Key Concepts

Concept	Definition	Example
Exponent	Indicates repeated multiplication	$3^4 = 3 imes 3 imes 3 imes 3$
Algebraic Expression	Combination of numbers, variables, and operations	$2x + 5$
Like Terms	Terms with same variable(s) and exponent(s)	$4x$, $7x$
Simplification	Combining like terms to reduce expression	$2a + 5a = 7a$

Additional info: These notes cover foundational algebraic concepts including exponents, algebraic expressions, translation of word problems, and simplification, which are essential for success in Beginning Algebra.