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Algebra 1 Unit 1: Expressions & real numbers

Algebra starts with expressions: combinations of numbers, variables and operations. Simplifying and evaluating them correctly is the base for everything else.

Covers: Order of operations, properties of real numbers, combining like terms, evaluating expressions

Study guide

Algebra starts with expressions: combinations of numbers, variables and operations. The order of operations decides how to evaluate them: parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right (PEMDAS). Be careful with negatives: (−3)² = 9 because −3 is the base, but −3² = −9 because the exponent applies before the negative.

The properties of real numbers explain why algebra moves are allowed. The commutative property lets you change order (a + b = b + a), the associative property lets you regroup, and the distributive property, a(b + c) = ab + ac, connects multiplication and addition. The identity and inverse properties explain why adding 0 or multiplying by 1 changes nothing, and why adding −a or multiplying by 1/a undoes a.

Like terms have the same variables raised to the same powers, such as 4x and −7x. Combine them by adding their coefficients; unlike terms like x and x² can't be combined. To evaluate an expression, substitute the given values (in parentheses, especially negatives) and follow the order of operations. Real numbers include integers, fractions, terminating and repeating decimals (rational numbers), and irrational numbers like √2 and π.

To simplify a square root, factor out the largest perfect square and take its root: √72 = √36·√2 = 6√2. Radicals multiply under one root sign, √a·√b = √(ab), and only like radicals (same radicand) can be added, as in 3√2 + 5√2 = 8√2. To remove a square root from a denominator, multiply the top and bottom by that root: 6/√3 = 6√3/3 = 2√3. The value stays the same because you are multiplying by a form of 1.

The sum or product of two rational numbers is always rational. A rational number plus an irrational number is irrational, and a nonzero rational number times an irrational number is irrational; two irrationals can give either result (√2·√2 = 2). The properties of equality say you may add, subtract, multiply or divide both sides of an equation by the same number (dividing only by a nonzero number) and keep the equation true. When a question asks which property justifies a step, name the operation applied to both sides, for example the addition property of equality.

Key ideas

• Order of operations: parentheses, exponents, multiply/divide left to right, add/subtract left to right.
• Like terms have the same variable raised to the same power.
• Distributive property: a(b + c) = ab + ac.
• When substituting a negative number, put it in parentheses.
• (−2)⁴ = 16 but −2⁴ = −16.

Formulas and rules

Common mistakes

• Distributing to only the first term in parentheses.
• Squaring a negative without parentheses.

Words to know

Order of operations

Parentheses, exponents, multiply/divide, add/subtract (PEMDAS).

Distributive property

a(b + c) = ab + ac

Like terms

Terms with the same variables and exponents.

Squaring a negative

(−3)² = 9: the parentheses make −3 the base.

Exponent before negation

−3² = −(3²) = −9: the exponent applies first.

Commutative property

Order doesn't matter: a + b = b + a and ab = ba.

Associative property

Grouping doesn't matter: (a + b) + c = a + (b + c).

Additive inverse

The opposite of a number; a + (−a) = 0.

Multiplicative inverse (reciprocal)

a · (1/a) = 1

Rational number

A number that can be written as a fraction of integers.

Irrational number

A number like √2 or π whose decimal never ends or repeats.

Coefficient

The number multiplying a variable, like 4 in 4x.

Evaluating an expression

Substituting values for variables and simplifying.

Sample practice questions

33 more questions like these, from level 1 to 5, are in the practice set.