Example Card: Using the Product Rule
Apply the Product Rule for exponents in Grade 9 algebra: when multiplying powers with the same base, add the exponents—aᵐ·aⁿ=aᵐ⁺ⁿ—to simplify exponential expressions quickly.
Key Concepts
When two quantities vary inversely, their product is always the same. Let's use this powerful rule to find a missing value. This example demonstrates one of the key ideas in this lesson, using the product rule.
Example Problem If $y$ varies inversely as $x$, and $y = 5$ when $x = 8$, find $x$ when $y = 10$.
Step by Step 1. Use the product rule for inverse variation, which states that for any two pairs $(x 1, y 1)$ and $(x 2, y 2)$, their products are equal: $x 1y 1 = x 2y 2$. 2. Substitute the given values into the formula. Let $x 1 = 8$, $y 1 = 5$, and $y 2 = 10$. The equation becomes: $(8)(5) = x 2(10)$. 3. Calculate the product on the left side of the equation: $40 = 10x 2$. 4. To find $x 2$, divide both sides by $10$: $\frac{40}{10} = x 2$. 5. The result is $x 2 = 4$. So, when $y = 10$, the value of $x$ is $4$.
Common Questions
What is the Product Rule for exponents?
The Product Rule states that when multiplying two powers with the same base, you keep the base and add the exponents: aᵐ · aⁿ = aᵐ⁺ⁿ. For example, x³ · x⁵ = x⁸.
How do you apply the Product Rule to expressions with coefficients?
Multiply the coefficients separately, then add the exponents of like bases. For (3x²)(4x⁵), multiply coefficients: 3 × 4 = 12, and add exponents: x² × x⁵ = x⁷. Result: 12x⁷.
Can the Product Rule be applied to different bases?
No. The Product Rule only applies when the bases are identical. x³ · y⁵ cannot be simplified using this rule because x and y are different bases. The rule requires aᵐ · aⁿ — the same letter a both times.