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...

Powers of Products and Quotients Übung

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  • Determine the missing term.

    Tipps

    An example using the power of products rule:

    An example using the power of quotients rule:

    If you raise a power to a power, you can write it as a product:

    Lösung

    Let's start with the power of products rule:

    $(a\times b)^m=a^m\times b^m$.

    This means that you have to raise each factor to the same power.

    To raise a power to a power, $(a^m)^p=a^{m\times p}$, keep the basis and multiply the exponents.

    Let's look at the power of quotients rule:

    $\left(\frac ab\right)^m=\frac{a^m}{b^m}$.

    Here we raise the numerator as well as the denominator by the power.

  • Decide which example belongs to which rule.

    Tipps

    Think about the meaning of the variables in each rule.

    Check if the rules are correct.

    An example of the power of product rule with $a=3$, $b=10$, and $m=2$:

    Lösung

    $\begin{array}{lrcl} \text{product}&(a\times b)^m&=&a^m\times b^m\\ \text{power}&(a^m)^p&=&a^{m\times p}\\ \text{quotient}&\left(\frac ab\right)^m&=&\frac{a^{\large m}}{b^{\large m}} \end{array} $

    Let's check each of the rules with an example.

    The power of product rule with $a=2$, $b=5$ and $m=3$:

    $\begin{array}{rclll} (2\times 5)^3&=&(2\times 5)\times(2\times 5)\times(2\times 5)&|&\text{commutative property}\\ &=&2\times 2\times 2\times 5\times 5\times 5&|&\text{definition of powers}\\ &=&2^3\times 5^3 \end{array}$

    The power of power rule with $a=2$, $m=3$ and $p=4$:

    $\begin{array}{rclll} (2^3)^4&=&(2^3)(2^3)(2^3)(2^3)&|&\text{add the exponents}\\ &=&2^{(3+3+3+3)}\\ &=&2^{12}=2^{3\times 4} \end{array}$

    the power of quotient rule with $a=2$, $b=5$ and $m=3$:

    $\begin{array}{rclll} \left(\frac25\right)^3&=&\left(\frac25\right)\left(\frac25\right)\left(\frac25\right)\\ &=&\frac{2\times 2 \times2}{5\times 5 \times 5}\\ &=&\frac{2^{\large 3}}{5^{\large 3}} \end{array}$

  • Prove that $(2\times 5)^3 = 2^5\times 3^5$ using the power of products rule.

    Tipps

    A power is shorthand for a product:

    Multiplication is commutative; i.e. $a\times b= b\times a$.

    Remember the power of products rule:

    Lösung

    Let's have a look at the power of products rule:

    $(a\times b)^m=a^m\times b^m$.

    Plugging in $a=2$, $b=5$, and $m=3$, we get that

    $(2\times 5)^3=(2\times 5)\times(2\times 5)\times(2\times 5)$.

    As multiplication is commutative, we can change the order of multiplication:

    $(2\times 5)^3=2\times 2\times 2\times 5\times 5\times 5$.

    We know that $2\times 2\times 2=2^3$ and that $5\times 5\times 5=5^3$, giving us $(2\times 5)^3=2^3\times 5^3$.

  • Rewrite each term using the power of a product, power, and quotient rules.

    Tipps

    To raise a product just raise each factor to the same exponent.

    To raise a quotient raise the numerator as well as the denominator to the same exponent.

    To raise a power keep the basis and multiply the exponents.

    Lösung

    We have the following equalities:

    • $\left(\frac23\right)^7=\frac{2^{\large 7}}{3^{\large 7}}$
    • $(3^3)^3=3^{3\times 3}=3^9$
    • $\left(\frac 56\right)^6=\frac{5^{\large 6}}{6^{\large 6}}$
    • $(3x)^4=3^4\times x^4$
    • $(xy)^5=x^5\times y^5$
    • $(2^x)^2=2^{x\times 2}=2^{2x}$
  • Complete the following examples.

    Tipps

    Remember the rules:

    An example of the power of a power rule with $a=4$ and $m=p=2$:

    Lösung

    Using the rules,

    $\begin{array}{lrcl} \text{product}&(a\times b)^m&=&a^m\times b^m\\ \text{power}&(a^m)^p&=&a^{m\times p}\\ \text{quotient}&\left(\frac ab\right)^m&=&\frac{a^{\large m}}{b^{\large m}} \end{array} $

    we can compute the concrete examples Charlotte needs to solve in her game:

    • $(4\times 3)^2=4^2\times 3^2$
    • $(4^3)^2=4^{3\times 2}=4^6$
    • $\left(\frac 43\right)^2=\frac{4^{\large 2}}{3^{\large 2}}$
    • $(3\times 2)^4=3^4\times 2^4$
  • Determine which terms are equal.

    Tipps

    Remember the power of a product rule:

    $(a\times b)^m=a^m\times b^m$.

    An example of the power of a quotient rule:

    To raise a power just multiply the exponents.

    Lösung

    Let's have a look at some examples with constants as well as variables.

    Using the power of a product rule, $(a\times b)^m=a^m\times b^m$, we have

    • $50^x=(5\times 10)^x$
    • $(2\times x)^2=2^2\times x^2=4x^2$
    Using the power of a quotient rule, $\left(\frac ab\right)^m=\frac{a^{\large m}}{b^{\large m}}$, we have
    • $0.5^x=\left(\frac48\right)^x=\frac{4^{\large x}}{8^{\large x}}$
    • $\frac{10^{\large 2}}{5^{\large 2}}=\left(\frac{10}5\right)^2=2^2=4$
    Using the power of power rule, $(a^m)^p=a^{m\times p}$, we have
    • $(x^2)^3=x^{2\times 3}=x^6$
    • $(2^3)^2=2^{3\times 2}=2^6$