As you saw in the video, there is another rule, called the The Quotient of Powers Property, that will aid in dividing exponential expressions correctly. In this example, we divide two expressions, each with the same base, and each with an exponent. Once you've read through the explanation, try the practice problems.a
Explanation
Practice #1
Practice #2
Read through the examples below. This is another explanation, similar to the video, showing expanded notation. When finished, click on the other tabs to try some practice questions on your own.
| \(\large\mathsf{ \frac{3^{5}}{3^{3}} = ? }\) \(\normalsize\mathsf{ \frac {3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}{3 \cdot 3 \cdot 3} }\) |
By expanding this expression, we can see how this property works. |
| \(\normalsize\mathsf{ \frac{3}{3} \cdot \frac{3}{3} \cdot \frac{3}{3} \cdot \frac{3}{1} \cdot \frac{3}{1} }\) \(\normalsize\mathsf{ 3 \cdot 3 = 3^{2} }\) |
We can now simplify. Three sets of \(\normalsize\mathsf{ \frac{3}{3} }\) will cancel. This leaves two sets of \(\normalsize\mathsf{ \frac{3}{1} }\), which can then be written as \(\normalsize\mathsf{ 3^{2} }\) |
| \(\large\mathsf{ \frac{3^{5}}{3^{3}} = 3^{5-3} = 3^{2} }\) | Here's the same expression using the Quotient of Powers Property. |
What is the missing exponent?
\(\mathsf{ \frac {5^{8}}{5^{2}} = 5^{?} }\)
The exponent should be 6. When dividing terms with exponents, just subtract the exponents. The exponents in this quotient are 8 - 2 = 6.
What is the missing exponent?
\(\large\mathsf{ \frac {x^{4}}{x^{2}} = x^{?} }\)
The correct exponent is 2. 4 - 2 = 2. The base must be the same. In this case the base is "x."
The quotient of powers property allows us to subtract the exponents when the
bases are identical. This makes for some quick division!
The Quotient of Powers Property
| \(\large\mathsf{ \frac{a^{n}}{a^{m}} = a^{n-m} }\) | To find the quotient of two numbers with exponents, just subtract the exponents. They must have the same base. |
\(\large\mathsf{ \frac{7^{8}}{7^{5}} = 7^{8-5} = 7^{3} }\) \(\large\mathsf{ \frac{5^{11}}{5^1} = 5^{11-1} = 5^{10} }\)
