0.7Unit 0 Review
Review all six lessons of Unit 0 Foundations — variables, order of operations, real numbers, properties of operations, exponents, and scientific notation.
Why This Matters
A strong foundation in these skills makes every future unit faster and less stressful. Students who review Unit 0 thoroughly consistently perform better on equations, functions, and graphing in the units ahead.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How do the six foundational skills of Unit 0 connect to each other, and how will they support every unit in Algebra 1?
Unit 0 Overview
Unit 0 builds the foundation for all of Algebra 1. In this unit you learned six essential topics. This review chapter brings them all together so you can check your understanding before the unit test.
Vocabulary Review
Variable
A letter that represents an unknown or changing value (e.g., x, y, n).
Algebraic Expression
A combination of variables, numbers, and operations — no equals sign.
PEMDAS
Order of operations: Parentheses, Exponents, Multiply/Divide, Add/Subtract.
Rational Number
Any number that can be written as a fraction p/q where q ≠ 0.
Irrational Number
A number that cannot be written as a fraction; its decimal never repeats (e.g., π, √2).
Commutative Property
Order does not change the result: a + b = b + a and a × b = b × a.
Distributive Property
a(b + c) = ab + ac — multiply the outside term by each term inside.
Exponent
Tells how many times to multiply the base by itself: 3⁴ = 3 × 3 × 3 × 3.
Scientific Notation
A number written as a × 10ⁿ where 1 ≤ a < 10.
Evaluate
Substitute a value for a variable and simplify to find the answer.
Key Ideas Summary
A variable stands for an unknown. Evaluate expressions by substituting and simplifying.
Always follow PEMDAS. Work left to right for × ÷ and for + −.
Real numbers include rationals and irrationals. Integers include negatives, zero, and positives.
Use commutative, associative, and distributive properties to simplify expressions.
Product rule: xᵃ · xᵇ = xᵃ⁺ᵇ. Quotient rule: xᵃ ÷ xᵇ = xᵃ⁻ᵇ. Power rule: (xᵃ)ᵇ = xᵃᵇ.
Move the decimal to get 1 ≤ a < 10, then count moves for the power of 10.
Worked Examples
Evaluate 3x² − 2x + 5 when x = 4.
Substitute x = 4: 3(4)² − 2(4) + 5
Exponent first: 3(16) − 2(4) + 5
Multiply: 48 − 8 + 5
Left to right: 40 + 5 = 45
Simplify: 5 + 3² × (8 − 6) ÷ 6
Parentheses: 8 − 6 = 2 → 5 + 3² × 2 ÷ 6
Exponent: 3² = 9 → 5 + 9 × 2 ÷ 6
Multiply/Divide left to right: 9 × 2 = 18 → 18 ÷ 6 = 3
Add: 5 + 3 = 8
Classify −7 in as many sets as possible.
Is it a natural number? No (negatives excluded).
Is it a whole number? No.
Is it an integer? Yes — integers include negatives.
Is it rational? Yes — it can be written as −7/1.
Is it real? Yes — all rationals are real.
Simplify: 4(3x − 5) + 2x
Distribute 4: 12x − 20 + 2x
Combine like terms: (12x + 2x) − 20
14x − 20
Simplify (2x³)⁴, then write the coefficient in scientific notation.
Power rule — raise each factor to the 4th power: 2⁴ · (x³)⁴
2⁴ = 16; (x³)⁴ = x¹² → 16x¹²
Write the coefficient 16 in scientific notation: 1.6 × 10¹
Final: 1.6 × 10¹ · x¹²
Common Mistakes
Skipping order of operations when substituting — e.g., computing 2a − b as 2(a − b).
Substitute first, then apply PEMDAS: multiply before subtracting.
Applying exponent rules to different bases — e.g., x² · y³ = (xy)⁵.
Exponent rules (product, quotient, power) only apply when the bases are identical.
Forgetting to distribute a negative sign when expanding — e.g., −(3x − 2) = −3x − 2.
Distribute the negative to every term: −(3x − 2) = −3x + 2.
Misclassifying numbers — e.g., thinking √9 is irrational because it has a radical sign.
√9 = 3, which is a whole number. Always simplify before classifying.
Guided Practice
Guided Practice Video: Unit 0 Review
Watch the guided practice walkthrough for the Unit 0 review, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Answers are in the Answer Key section.
Evaluate 2a − b when a = 5, b = 3.
Hint: Substitute first, then subtract.
Simplify: 24 ÷ (6 − 2) + 3²
Hint: Parentheses first, then exponent.
Classify √5 in as many real number sets as possible.
Hint: Can √5 be written as a fraction?
Simplify using the distributive property: 3(2x + 7)
Hint: Multiply 3 by each term inside.
Simplify: x⁵ · x³
Hint: Add the exponents — same base.
Write 0.000047 in scientific notation.
Hint: Move the decimal right until 1 ≤ a < 10.
Practice Questions
Interactive Practice — 5 Questions
Evaluate 2x² − 3x + 1 when x = 4.
Simplify: 3 + 2² × (5 − 2) ÷ 6
Which set does √2 belong to?
Simplify: 3(4x − 2) − 5x
Simplify (3x²)³ ÷ x², then identify the correct scientific notation for the coefficient.
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Evaluate 4x² − 3x + 1 when x = 2.
Simplify: 2 + 3² × (4 − 1) ÷ 9
Classify 0 in all applicable number sets.
Expand and simplify: 3(2x − 4) + 5x
Simplify: (2x³)² · x⁴, then write 4 × 10⁻² in standard form.
Math Tips
Unit 0 is the foundation for all of Algebra 1. If any topic feels shaky, review that chapter before moving on — a weak foundation makes every future unit harder.
When evaluating expressions, always write the substitution step first. When simplifying with PEMDAS, rewrite after each step. These habits prevent most careless errors.
Exponent rules and scientific notation are heavily tested on the SAT/ACT. Make sure you can apply all five exponent rules quickly and accurately.
The Distributive Property is the most-used algebraic tool in Algebra 1. You will apply it in every unit — equations, inequalities, functions, and polynomials all rely on it.