Combine like terms to simplify expressions

Key idea

You can combine only like terms. To combine them, add or subtract the coefficients and keep the same variable part. A helpful trick is to rewrite subtraction as adding a negative: \[ a-b=a+(-b). \]

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Summary

• Only combine like terms.
• Watch signs: \(a-b=a+(-b)\).
• Keep the variable part the same when you combine coefficients.

Expand brackets using the distributive property

Key idea

The distributive property says you multiply the number outside the parentheses by every term inside: \[ k(a+b)=ka+kb \quad \text{and} \quad k(a-b)=ka-kb. \] A common pattern is: expand first, then combine like terms.

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Summary

• Distribute to every term inside parentheses.
• After expanding, combine like terms to finish simplifying.

Simplify powers using exponent rules

Key idea

Two rules are especially useful when simplifying algebraic expressions: \[ (x^m)^n=x^{mn} \quad\text{and}\quad x^0=1 \text{ (for } x\ne 0\text{)}. \] Also, when a product is squared, each factor is squared: \[ (ab)^2=a^2b^2. \]

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Summary

• Powers of powers: \((x^m)^n=x^{mn}\).
• Zero exponent: \(x^0=1\) for \(x\ne 0\).
• Square a product by squaring each factor: \((ab)^2=a^2b^2\).

Simplify algebraic fractions (rational expressions)

Key idea

To simplify an algebraic fraction, factor and cancel common factors (not common terms). Also remember: any value that makes the denominator zero is not allowed.

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Summary

• Factor and cancel common factors, not terms.
• Always state/remember restrictions: the denominator cannot be \(0\).

Factor using the greatest common factor (GCF)

Key idea

To factor by the greatest common factor, find the largest factor shared by every term, then rewrite the expression as a product. Factoring is the reverse of distributing.

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Summary

• Factoring by GCF rewrites a sum as a product.
• Factoring is the reverse of distributing, so you can always check by expanding.

Multi-step simplification (a reliable order)

Key idea

When an expression has several features (parentheses, negatives, powers), use a consistent process:

• 1) Simplify powers (like \((x^2)^2\)).
• 2) Expand parentheses (distribute).
• 3) Combine like terms and simplify constants.
• 4) Check signs (especially with subtraction and double negatives).

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Summary

• Use a consistent order: powers → distribute → combine like terms → check signs.
• Double negatives matter: \(-(-x)=x\).

Why simplifying expressions matters

Where you use algebraic simplification

• Solving equations: simplifying first makes it easier to isolate a variable.
• Functions and graphs: simpler expressions are easier to evaluate and compare.
• Science and modeling: expressions describe relationships (speed, area, cost, growth).
• Checking work: a simpler form helps you spot mistakes and verify equivalence.

Worked example: simplifying a formula

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Fun fact (a little history)

• Word origin: The word “algebra” comes from al-jabr, associated with the mathematician al-Khwarizmi.
• Big idea: Simplifying expressions is like simplifying a sentence: it keeps the meaning the same, but makes it clearer to work with.

Final recap

• Combine like terms by adding/subtracting coefficients.
• Expand brackets using the distributive property.
• Use exponent rules, especially \((x^m)^n=x^{mn}\) and \(x^0=1\) (for \(x\ne 0\)).
• Simplify algebraic fractions by canceling common factors (and keep restrictions in mind).
• Factor expressions by pulling out the greatest common factor.