Exponent Laws — Combined Practice

Master all exponent laws and their applications. Math 10-C Alberta mathematics curriculum.

Lesson 2.6 of Number Systems & Operations in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• The Exponent Laws — Complete Reference. A full summary of every exponent law in one place, plus a worked strategy example showing why you should always simplify before substituting.
• Simplifying with Product, Quotient, and Power Laws. Practice multiplying and dividing expressions with exponents, then apply the power rule when an entire expression is raised to a power.
• Fractional Exponents Combined with Exponent Laws. Extend the exponent laws to fractional exponents and practice rewriting expressions with positive exponents.
• Application — Cone Problem. Use a rational exponent and the cone volume formula to find unknown dimensions of a cone, connecting algebra to real-world geometry.

Inside the lesson — a free preview

All Eight Exponent Laws

Why a Complete Reference?. You have seen most of these laws before in earlier lessons. This tab collects all eight laws in one place so you can see how they fit together before applying them in combination.

An expression is considered fully simplified when: there are no brackets remaining, no negative exponents, and no zero exponents. Any negative exponents must be rewritten as positive exponents using the reciprocal rule.

Strategy: Simplify First, Then Substitute

Why Simplify Before Substituting?. When an expression contains variables and you need to evaluate it at specific values, always simplify the expression first using the exponent laws, then substitute.

This avoids messy arithmetic with large numbers. If you substitute first into something like (a^6b^9 ÷ a^5b^8)^(-2), you would need to compute (-3)^6 = 729 and 2^9 = 512 before doing anything else — far harder than simplifying first.

Worked Example — Evaluate (a^6b^9 ÷ a^5b^8)^(-2) when a = -3 and b = 2. Step 1 — Simplify the expression using exponent laws.

Apply the quotient rule inside the brackets:

(a^6b^9)/(a^5b^8) = a^(6-5) · b^(9-8) = a^1b^1 = ab

So the expression becomes:

(ab)^(-2)

Apply the negative exponent rule:

(1)/((ab)^2) = (1)/(a^2b^2)

Step 2 — Substitute a = -3 and b = 2.

(1)/((-3)^2 · 2^2) = (1)/(9 · 4) = (1)/(36)

Answer: (1)/(36)

Multiplying Expressions

The Rule for Multiplying. When multiplying two expressions with the same variable base, add the exponents on matching bases and multiply the coefficients together.

Example 1 — Simplify (3a^3b^(-2))(15a^2b^5). Step 1 — Multiply the coefficients.

3 × 15 = 45

Step 2 — Add the exponents on each base.

Base a: 3 + 2 = 5, so a^5

Base b: -2 + 5 = 3, so b^3

Answer: 45a^5b^3

Example 2 — Simplify (-2m^2n)(4mn^(-1)). Step 1 — Multiply the coefficients.

-2 × 4 = -8

Step 2 — Add the exponents on each base.

Base m: 2 + 1 = 3, so m^3

Base n: 1 + (-1) = 0, and n^0 = 1

Step 3 — Remove the zero exponent.

-8m^3 · 1 = -8m^3

Answer: -8m^3

Example 3 — Simplify m^4n^(-2) · m^2n^3. Step 1 — Add exponents on each base.

Base m: 4 + 2 = 6, so m^6

Base n: -2 + 3 = 1, so n^1 = n

Answer: m^6n

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