2.03 Product of powers property
Product of powers property
When multiplying a number by itself repeatedly, we are able to use exponent notation to write the expression more simply. Here we are going to look at a rule that allows us to simplify products that involve the product of powers.
Consider the expression a^{5} \times a^{3}. Notice that the terms share like bases.
Let's think about what this would look like if we distributed the expression:
We can see that there are eight a's being multiplied together, and notice that 8 is the sum of the powers in the original expression.
We can write this in exponential form as a^8, where a is the base and 8 is the power.
So, in our example above, \begin{aligned}a^{5}\times a^{3} &= a^{5+3}\\&=a^{8}\end{aligned}
We can avoid having to write each expression in expanded form by using the product of powers property. It states that when multiplying two powers with the same base, we add the powers.
For any base number a, and any numbers m and n as powers, a^{m} \times a^{n}=a^{m+n}.
That is, when multiplying terms with a common base:
Keep the same base
Find the sum of the exponents
Examples
Example 1
Fill in the blank to make the following eqution true: 4^{2}\times 4^{⬚} = 4^{2 + 3}.
Example 2
Simplify the following, giving your answer in exponential form: 2^{2}\times 2^{3}.
Example 3
Simplify the following, giving your answer in exponential form: 8^{2} \times 8^{7} + 3^{3} \times 3^{2}.
For any base number a, and any numbers m and n as powers, a^{m} \times a^{n}=a^{m+n}
When multiplying terms with like bases, we add the powers.