Powers of Products and Quotients

Powers of Products and Quotients exercise

• Determine the missing term.

  Hints

  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:

  Solution

  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.

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

  Hints

  A power is shorthand for a product:

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

  Remember the power of products rule:

  Solution

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

• Complete the following examples.

  Hints

  Remember the rules:

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

  Solution

  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.

  Hints

  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.

  Solution

  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

  • $(x^2)^3=x^{2\times 3}=x^6$
  • $(2^3)^2=2^{3\times 2}=2^6$

• Decide which example belongs to which rule.

  Hints

  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 $m=2$:

  Solution

  $\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 $m=3$:

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

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

  Hints

  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.

  Solution

  We have the following equalities:

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