Powers of Products and Quotients 05:20 minutes

Video Transcript

Transcript Powers of Products and Quotients

Charlotte is playing a new game, Pow Pow Powers, but the kids call it 'Triple P'. She is stuck on a level at the moment in this level, she has to complete the rules for raising products and quotients to a power. Surely, her knowledge of Powers of Products and Quotients will help her dance her way to the top of the leaderboard.

Power of Products and Quotients Rules

Charlotte has to complete the rules for raising a product, a quotient, and a power to a power in order to achieve gaming immortality. She starts with the power of a product. The variables are coming faster now
"'a'! Times! 'b'! all to the 'm' power! Equals! 'a'! To the 'm' power! Times! 'b' to the 'm' power!"
Next up is raising a power to a power! The variables just keep coming!
"'a'! To the 'm' power! all to the 'p' power! Equals! 'a'! To the 'm'! Times 'p'! Power!"
Last one! Raising a quotient to a power! Can Charlotte keep up?!
"'a'! Divided by! All to the 'm' power! Equals! 'a'! To the 'm'! Divided by! 'b'! to the 'm'! Power!"
Let's look at an example of the rules Charlotte needed to know for the game with real numbers.

Raising Products to a Power

The first rule was for raising products to a power. [a(b)]ᵐ = aᵐ(bᵐ) Let's plug in 2 for 'a', 5 for 'b' and 3 for 'm'. Is [2(5)]³ the same as (2³)(5³)? Let's think of [2(5)]³ as (2)(5) (2)(5) (2)(5).
According to the Commutative Property of Multiplication, you can change the order of the factors to be (2)(2)(2)(5)(5)(5). To simplify this, you can write (2³)(5³). This means that [2(5)]³ is the same as (2³)(5³).

Raising a Power to a Power

But, how do you raise a power to a power? This rule states that (aᵐ)ᵖ = aᵐ⁽ᵖ⁾. Let's let 'a' be 2 again. 'm' is still 3. 'p' will be 4. Is (2³)⁴ the same as 2¹²? Let's imagine multiplying 2³ by itself four times. Since the bases are the same, we can add the exponents together. There are four exponents with the value 3, so we can write the expression as 2⁽³⁾⁽⁴⁾. To simplify this, we can multiply the exponents, giving us 2¹².

Raising Quotients to a Power

For her last feat, Charlotte raise quotients to a power. This rule states that (a/b)ᵐ = aᵐ / bᵐ. What does this look like when we use numbers? For this, 'a' will be 2 again, 'b' will be 5 and 'm' will be 3. Which brings us to our question: what is (2/5)³ equal to?
According to the Commutative Property of Multiplication, you can rewrite this expression as (2/5)(2/5)(2/5). For the numerator, we can write (2)(2)(2) and for the denominator, (5)(5)(5). Simplifying this gives us (2)³ / (5)³.

Let's see how Charlotte's getting along in the game. She did it! First place! A NEW #1 and bragging rights! But just as Charlotte goes to enter her name as the champion of the household...