Unit 1 · Core unit

Real Numbers and Algebraic Expressions

The rules of arithmetic, restated so they still hold once letters replace numbers. Exponent rules and order of operations do most of the work here and are assumed silently by every later unit.

What a strong answer looks like

A strong answer applies one rule at a time and names it, rather than collapsing several steps and hoping.

Practice this unit Course overview

Topics in this unit

Order of Operations and Sign Rules

Know

Operations resolve in a fixed order, and a sign belongs to the term it precedes. Most arithmetic errors in algebra are sign errors rather than genuine misunderstandings.

Apply

Resolve brackets, then exponents, then multiplication and division, then addition and subtraction, writing each step.

Watch out

Distributing a negative across only the first term inside a bracket.

Study move

Expand three expressions with a negative outside the bracket and check every sign.

Exponent Rules

Know

Multiplying like bases adds exponents, dividing subtracts, and a power of a power multiplies. A rational exponent is a root: the denominator is the index.

Apply

Apply the outermost exponent first, then simplify each factor separately.

Watch out

Adding exponents when raising a power to a power, which should multiply them.

Study move

Simplify an expression combining a negative and a fractional exponent, one rule per line.

Rational and Irrational Numbers

Know

A rational number is a ratio of integers. Products of irrationals can be rational, so the classification depends on the simplified value, not its appearance.

Apply

Simplify first, then classify.

Watch out

Assuming any expression containing a root is irrational.

Study move

Classify four expressions, at least one of which simplifies to an integer.

Emphasized in this unit

• Writing one rule per line
• Tracking signs deliberately
• Simplifying before classifying

Varies by course

Scientific notation. Emphasised in courses taken alongside science.
Absolute value equations. Introduced here in some sequences.

Mastery checklist

• Expand an expression with a leading negative correctly.
• Simplify combined exponent rules.
• Classify a number after simplifying it.

Check yourself

• Why do most expansion errors turn out to be sign errors?
• What does the denominator of a rational exponent mean?

Modeling drill

Simplify an expression combining a fractional exponent, a negative exponent, and a division, naming the rule used at each step.

Order of operationsDistributionExponent ruleRational exponentIrrational

Unit 2: Linear Equations and Inequalities