Division with Exponents

Division with Exponents exercise

• Explain the rule for dividing exponents.

  Hints

  Notice in this example, the matching base is the variable $n$.

  This stays the same.

  Another way to understand dividing exponents is to expand each exponent. Then it will make it clear that subtraction is used to find the final exponent.

  Solution

  When dividing exponents that have the same base, the rule is keep the base the same and subtract the exponents.

  For example, $\dfrac{3^7}{3^3}$, the base is $\bf{3}$, so this stays the same. To find the quotient's exponent, subtract the exponents: $\bf{7-3}$$=4$. The quotient is equal to $\bf{3^4}$.

• Use expansion to show the expression in exponential notation.

  Hints

  To expand an exponent, the base is multiplied by itself the number of times the value of the exponent is.

  For example, $4^3$ = $4 \cdot 4 \cdot 4$.

  $\dfrac{5^6}{5^4}$

  For the numerator, the base is $5$, and there are $6$ of them being multiplied. So, $5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$.

  What is the base of the denominator, and how many times is that number being multiplied by itself?

  Solution

  $\dfrac{5^6}{5^4}$

  The numerator has a base of $5$, and since the exponent is $6$, the $5$ is expanded as: $5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5$.

  The denominator has a base of $5$, and the exponent is $4$, so the $5$ is expanded as: $5\cdot 5\cdot 5\cdot 5$

  $\dfrac{5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5}{5\cdot 5\cdot 5\cdot 5}$

• Divide the exponents using the rule.

  Hints

  The Law of Exponents for dividing says that if you divide two numbers with the same base, you just subtract the smaller exponent from the bigger one and keep the base the same. For example, $a^m \div a^n = a^{m-n}$ when $a$ is a number and $m$ and $n$ are the exponents.

  Here is an example to help you.

  $\dfrac{2^7}{2^6}$

  $2^{7-6}$

  $2^1=2$

  Solution

  $\dfrac{5^9}{5^3}$

  $5^{9-3}$

  $5^6$

• Apply the law of exponents to evaluate expressions in exponential notation.

  Hints

  The Law of Exponents for dividing exponents states that when you divide two numbers with the same base, you keep the base the same and subtract the exponents.

  If it helps you to better understand the rule, first expand the expression written in exponential notation and then cross off pairs.

  Solution

  • $\frac{4^7}{4^3} = 4^{7-3} = 4^4$
  • $\frac{4^4}{4} = 4^{4-1} = 4^3$
  • $\frac{3^6}{3^2} = 3^{6-2} = 3^4$
  • $\frac{4^8}{4^2} = 4^{8-2} = 4^6$

• Apply the rule for dividing exponents.

  Hints

  The Law of Exponents for dividing exponents states that when you divide two numbers with the same base, you keep the base the same and subtract the exponents.

  The base in this example is $3$, and then the exponents are subtracted, $6-5$.

  Solution

  The base is $2$.

  The exponents will be subtracted $8-2$.

  The final solution will be $2^6$.

• Find the solution by applying the laws of exponents.

  Hints

  When you type in your answer, use the $x^a$ button on the toolbar to create an exponent.

  The Law of Exponents for dividing says that if you divide two numbers with the same base, you just subtract the smaller exponent from the bigger one and keep the base the same.

  For example, $a^m \div a^n = a^{m-n}$ when $a$ is a number and $m$ and $n$ are the exponents.

  Solution

  $\begin{array}{l}\frac{5^5}{5^2}=5^{5-2}=5^3\\ \\ \frac{8^2}{8}=8^{2-1}=8\\ \\ \frac{9^7}{9^2}=9^{7-2}=9^5\\ \\ \frac{x^6}{x^4}=x^{6-4}=x^2\end{array}$