Multiplying and Dividing Monomials

MATH REVIEW & SEATWORK

MULTIPLYING & DIVIDING MONOMIALS

Master the rules of exponents! Read the lessons carefully, study the examples, then take the 5-item seatwork!

Rule 1: Product Rule (Multiplication)

When multiplying monomials with the same base, multiply the coefficients (numbers), then ADD the exponents of the same variables.

Example 1: Basic Multiplication

Simplify: x³ · x⁴

Step 1: The bases are the same (both are 'x'). →

Step 2: Add the exponents: 3 + 4 = 7. →

Final Answer: x⁷

Example 2: With Coefficients

Simplify: (2a²)(3a)

Step 1: Multiply coefficients: 2 × 3 = 6. →

Step 2: Add exponents of 'a': 2 + 1 = 3. (Note: 'a' means 'a¹'). →

Final Answer: 6a³

Rule 2: Quotient Rule (Division)

When dividing monomials with the same base, divide the coefficients, then SUBTRACT the exponents (Top exponent - Bottom exponent).

Example 3: Basic Division

Simplify: x⁵x²

Step 1: Bases are the same. Keep the base 'x'. →

Step 2: Subtract exponents: 5 - 2 = 3. →

Final Answer: x³

Example 4: With Coefficients

Simplify: 12m⁴4m

Step 1: Divide coefficients: 12 ÷ 4 = 3. →

Step 2: Subtract exponents of 'm': 4 - 1 = 3. →

Final Answer: 3m³

Rule 3: Power of a Power Rule

When a monomial is raised to another power, raise the coefficient to that power, and MULTIPLY the exponents.

Example 5: Power Rule

Simplify: (2x³)²

Step 1: Square the coefficient: 2² = 4. →

Step 2: Multiply the exponents: 3 × 2 = 6. →

Final Answer: 4x⁶

Rule 4: Negative Exponent Rule

A negative exponent means the term wants to move to the other side of the fraction bar. Once it moves, the exponent becomes positive!

Example 6: Negative Exponents

Simplify: x⁻³

Step 1: 'x' has a negative exponent. It needs to move to the denominator. →

Step 2: Make the exponent positive. →

Final Answer: 1 / x³

Example 7: Division resulting in Negative Exponent

Simplify: a³a⁷

Step 1: Subtract exponents: 3 - 7 = -4. So we have a⁻⁴. →

Step 2: Move 'a' to the denominator to make the exponent positive. →

Final Answer: 1 / a⁴

Example 8: Putting it all together

Simplify: (2x²)³4x⁵

Step 1 (Numerator): Apply Power Rule. (2³)(x^(2×3)) = 8x⁶. →

Step 2 (Divide): Now we have 8x⁶4x⁵. Divide coefficients: 8 ÷ 4 = 2. →

Step 3 (Subtract): Subtract exponents: 6 - 5 = 1. →

Final Answer: 2x (or 2x¹)

Seatwork Time!

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