P.O.W.E.R Compendium (Lesson 1)

​Lesson 1: Product Rule

By: Group 3 Design Thinkers

Pre-Assessment

Please answer the questions Good luck!

Multiple Choice

Simplify: m⁸ x m²

• m¹⁰

• m⁶

• m¹⁶

• 2m¹⁰

Multiple Choice

3⁷ x 3 ³

• 3¹⁰

• 9¹⁰

• 3²¹

• 6¹⁰

Multiple Choice

Simplify: (x⁸ ) (x²)

• x⁸

• x¹⁵

• x¹⁰

• 2x¹⁰

Multiple Choice

Simplify: (3y⁴ ) (2y⁷)

• 5y¹¹

• 6y²⁸

• 6y¹¹

• 5y²⁸

Multiple Choice

If a^m x a⁴ = a¹² , find the value of m.

• 8

• 16

• 48

• 3

Objective

by the end of this lesson, students will be able to correctly apply the product rule of exponents a^m x aⁿ = a ^{m + n} to simplify algebraic expressions involving multiplication of exponential terms with the same base.

Work-it Out

The product rule of exponents is a fundamental rule used to simplify expressions involving multiplication of exponential terms with the same base. The rule states: a^m x aⁿ = a ^{m + n}

where a is the base (a nonzero number or variable), and m and n are exponents (any real numbers). This means that when multiplying two expressions that have the same base, you keep the base the same and add the exponents.

Try this!

Example 1.

Simplify: (2³ ) (2⁵)
Using the product rule, add the exponents
Answer 2³⁺⁵ = 2⁸

Work-it Out

Example 2:
Simplify: (3a²)(4a³)
Multiply coefficients: 3 × 4 = 12
add exponents on a : a ²⁺³ = a
_{Result : 12 a}⁵

Example 3.

Simplify: (5m³ n²) (2m⁴ n)
Multiply coefficients: 5×2=10
Add exponents for m: 3+4=7
Add exponents for n: 2+1=3
Result: 10m⁷ n³

Exercises

Now its your turn! Do these exercises

Multiple Choice

Simplify the expression:

4a³ x 2a⁵

• 6a⁸

• 8a⁸

• 8a¹⁵

• 6a¹⁵

Multiple Choice

3b⁷ x b⁴

• 3b ²⁸

• 4b ²⁸

• 3b ¹¹

• 4b ¹¹

Multiple Choice

Simplify the expression:

(6m²n³) (3m⁴n)

• 18m⁶n³

• 9m⁶n⁴

• 18m⁸n³

• 18m⁶n⁴

Recap

• Definition:
  The product rule of exponents states that when multiplying two exponential expressions with the same base, you keep the base and add the exponents:

  a^m x aⁿ = a ^{m + n}

• Key Points:

  • The bases must be the same for the rule to apply.

  • The exponents can be any real numbers, including positive, negative, or zero.

  • Multiplying coefficients is done separately from applying the exponent rule.

  • This rule simplifies multiplying powers and avoids expanding all the repeated multiplications.

Recap

• Additional notes

  • If the bases are different, the product rule does not apply this way.

  • The product rule is one of several laws of exponents helpful in simplifying algebraic expressions.