Sometimes, you need to combine expressions with exponents, and you'll need to handle more than one exponent at a time. There are "rules" in algebra to help you perform the right calculations. In this example, we multiply two expressions, each with the same base, and each with an exponent. Once you've read through the explanation, try the practice problems.
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.
| \(\mathsf{ 3^{2} \cdot 3^{5} }\) \(\mathsf{ (3 \cdot 3)(3 \cdot 3 \cdot 3 \cdot 3 \cdot 3) }\) \(\small\mathsf{ 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 3^{7} }\) |
By expanding this equation, we can count all of the 3's to find that there are seven 3's. |
| \(\mathsf{ 3^{2} \cdot 3^{5} = 3^{2+5} = 3^{7} }\) | We can take this same equation and use the Product of Powers Property to write the expression more simply and more quickly. The property tells us we can add exponents when we multiply numbers with identical bases. |
What is the missing exponent?
\(\mathsf{ 6^{2} \cdot 6^{7} = 6^{?}}\)
\(\mathsf{ 6^{2} \cdot 6^{7} = 6^{2+7} = 6^{9}}\)
What is the missing exponent?
\(\mathsf{ y^{4} \cdot y^{m} = y^{?}}\)
\(\mathsf{ y^{4} \cdot y^{m} = y^{4+m}}\)
The product of powers property allows us to add the exponents when the bases are identical. It's another shortcut worth remembering.
The Product of Powers Property
| \(\mathsf{ {a^{n}}{a^{m}} = a^{n+m} }\) | To find the product of two numbers with exponents, just add the exponents. They must have the same base. |
\(\mathsf{ 2^{7} \cdot 2^{2} = 2^{9} }\) \(\mathsf{ 7^{4} \cdot 7^{1} = 7^{5} }\)
