Exponents and Multiplication – Product of Powers Property

Exponents and Multiplication – Product of Powers Property exercise

• Find out the number of words Ethan has to write by using the Product of Powers Rule.

  Hints

  A power can also be represented with multiplication.

  For example, ${\large 2^4=2\times 2\times 2\times 2}$. So the base, ${\large 2}$, is written out as four factors that are multiplied.

  ${\large \begin{array}{rcl} a^n\times a^m&=&\underbrace{a\times\dots\times a}_{n\text{ times}}\times\underbrace{a\times\dots\times a}_{m\text{ times}}\\ &=&\underbrace{a\times\dots\times a}_{n+m\text{ times}}\\ &=&a^{n+m} \end{array}}$

  If you have $10$ to any positive power, write it as $1$ with as many zeros following as the exponent shows.

  Solution

  When Ethan got the message from his teacher, he thought that $10^3$ words per day for $10^2$ days would be completely doable. Poor Ethan.

  We have to multiply $10^3$ by $10^2$ to determine the total number of words.

  • $10^3$ stands for $10\times 10\times 10$.
  • $10^2$ stands for $10\times 10$.
  • $10^3\times 10^2=10\times 10\times 10\times 10\times 10$.

  To simplify this procedure, we use the Products of Powers Rule: If you have to multiply powers with the same base, just add the exponents.

  This gives us $10^3\times 10^2=10^{3+2}=10^5=100,000$.

  Ethan has to write $100,000$ words...like we said...poor Ethan.

• Determine how many words Ethan will be able to write in the next $16$ hours.

  Hints

  ${\Large \begin{array}{rcl} a^n\times a^m&=&\underbrace{a\times\dots\times a}_{n\text{ times}}\times\underbrace{a\times\dots\times a}_{m\text{ times}}\\ &=&\underbrace{a\times\dots\times a}_{n+m\text{ times}}\\ &=&a^{n+m} \end{array}}$

  Solution

  All we have to do here to multiply the two numbers, $2^{10}$ and $2^4$, is add the exponents:

  $2^{10}\times 2^4=2^{14}$

  $2^{14}$, or multiplying $2$ by itself fourteen times, is equal to $16,384$ words. This is the total number of words Ethan can write in $16$ hours.

• Using the Product of Powers Property, identify the base-exponent pair(s) that are factors of the displayed base-exponent.

  Hints

  $a=a^1$ for any base $a$.

  To use the Product of Powers Rule, the bases must be the same, but the exponents can differ.

  Solution

  We can only use the Product of Powers Property for exponents with the same base.

  $\begin{array}{rcl} a^n\times a^m&=&\underbrace{a\times\dots\times a}_{n\text{ times}}\times\underbrace{a\times\dots\times a}_{m\text{ times}}\\ &=&\underbrace{a\times\dots\times a}_{n+m\text{ times}}\\ &=&a^{n+m} \end{array}$

  It doesn't matter if the exponents differ.

  So we can simplify as follows:

  • ${\large x^2\times x^3=x^{2+3}=x^5}$
  • ${\large x\times x^3=x^{1+3}=x^4}$
  • ${\large x^5\times x^3=x^{5+3}=x^8}$
  • ${\large 2^2\times 2^{16}=2^{2+16}=2^{18}}$
  • ${\large 2^3\times 2^{16}=2^{3+16}=2^{19}}$
  • ${\large 2^7\times 2^{16}=2^{7+16}=2^{23}}$
  • ${\large 2\times 2^{16}=2^{1+16}=2^{17}}$
  • ${\large 3^2\times 3^2=3^{2+2}=3^4}$
  • ${\large 3^3\times 3^2=3^{3+2}=3^5}$
  • ${\large 3^5\times 3^2=3^{5+2}=3^7}$

• Find out how many animals Ethan has seen by using the Product of Powers Property.

  Hints

  ${\large 2=2^1}$

  How do you multiply two powers with the same base?

  Solution

  Here we have always the same base, $2$.

  We can apply the Product of Powers Property by adding the exponents of numbers with the same base to figure out the number of animals Ethan has seen sitting by the lake.

  Ducks

  $\large {2^2\times 2^2=2^{2+2}=2^4=16}$

  Ethan has seen $16$ ducks.

  Frogs

  $\large {2\times 2^4=2^1\times 2^4=2^{1+4}=2^5=32}$

  Ethan has also seen $32$ frogs.

  Fish

  $\large {2^2\times 2^5=2^{2+5}=2^7=128}$

  There are $128$ fish in the lake.

  Swans

  $\large {2^2\times 2=2^2\times 2^1=2^{2+1}=2^3=8}$

  $8$ swans passed by Ethan while he was relaxing.

• Describe how to multiply powers with the same base according to the Power of Products Property.

  Hints

  You can write each power as a product.

  Here is an example for the rule.

  Solution

  We've illustrated the Product of Powers Property below. You can use this property when multiplying numbers with the same base raised to an exponent.

  ${\large \begin{array}{rcl} a^n\times a^m&=&\underbrace{a\times\dots\times a}_{n\text{ times}}\times\underbrace{a\times\dots\times a}_{m\text{ times}}\\ &=&\underbrace{a\times\dots\times a}_{n+m\text{ times}}\\ &=&a^{n+m} \end{array}}$

  We can only apply this rule if we have two powers with the same base, for example $\large {a}$.

  You multiply two powers with the same base, $\large {a^n}$ and $\large {a^m}$, by adding the exponents $\large {a^{n+m}}$, while keeping the base the same.

  $~$

  Let's check this with a few examples:

  • $\large {2^2\times 2^3=(2\times 2)\times(2\times2\times2)=2^{2+3}=2^5}$
  • $\large {4^3\times 4^1=(4\times 4\times 4)\times(4)=4^{3+1}=4^4}$
  • $\large {5^2\times 5^4=(5\times 5)\times(5\times5\times5\times5)=5^{2+4}=5^6}$

• Simplify the given products of powers by using the Product of Powers Property.

  Hints

  If you have more than two factors, you can generalize the rule:

  ${\Large \begin{array}{rcl} 4^3\times 4^2\times 4^5&=&4^{3+2}\times 4^5\\ &=&4^5\times 4^5\\ &=&4^{5+5}\\ &=&4^{10}\\ &=&4^{3+2+5} \end{array}}$

  ${\Large a^n\times a^m=a^{n+m}}$.

  ${\Large a=a^1}$

  Solution

  Using the Product of Powers Rule, we can simplify the multiplication of powers with the same base.

  If we have more than one factor, we can still apply the rule: $\large a^n\times a^m\times a^k=a^{n+m+k}$.

  When applying the Product of Powers Property, we get:

  • ${\large 3^5\times 3\times 3^2=3^{5+1+2}=3^8}$
  • ${\large 2^{16}\times 2^4=2^{16+4}=2^{20}}$
  • ${\large 10^3\times 10^2\times 10^4\times 10=10^{3+2+4+1}=10^{10}}$
  • ${\large 5^5\times 5^5\times 5^5\times 5^5=5^{5+5+5+5}=5^{20}}$