5.01 Product rule
Ideas
Product rule
Exploration
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When a^3 is written in expanded form, how many a's are being multiplied?
When a^3 \cdot a^2 is written in expanded form, how many a's are being multiplied?
By counting the number of a's being multiplied in expanded form, what is a^3 \cdot a^2 in exponential form?
By counting the number of a's being multiplied in expanded form, what is a^4 \cdot a^5 in exponential form?
Is there a faster way to multiply terms with the same base?
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 of expressions with exponents.
Consider the expression a^{5} \cdot a^{3} Notice that the terms share like bases.
Let's think about what this would look like if we expanded the expression:
a^{5} \cdot a^{3} = (a\cdot a \cdot a \cdot a \cdot a)(a\cdot a\cdot a)
a^{5} \cdot a^{3} = a \cdot a \cdot a\cdot a\cdot a\cdot a\cdot a\cdot a
a^{5} \cdot a^{3} = a^8
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 the example above, \begin{aligned}a^{5}\cdot a^{3} &= a^{5+3}\\&=a^{8}\end{aligned}
We can avoid having to write each expression in expanded form by using the product rule. 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} \cdot a^{n}=a^{m+n}
That is, when multiplying terms with a common base:
Keep the same base
Find the sum of the exponents
When using the product rule for exponents, the coefficients are handled separately from the exponents. Let's take a look at an example.
3a^{2} \cdot 4a^{5}
Multiply the numerical coefficients: 3 \cdot 4 =12
Apply the product rule to the exponents: a^{2} \cdot a^{5}=a^{2+5}=a^7
The final result is:3a^{2} \cdot 4a^{5} = 12a^7
Examples
Example 1
Fill in the blank to make the equation true: b^{2}\cdot b^{⬚} = b^{2 + 3}
Example 2
Simplify m^{2} \cdot m^{7} + r^{3} \cdot r^{2}, giving your answer in exponential form.
Example 3
Simplify:
3^5 \cdot 3^9
c^{7} \cdot c^{6}
5d^{5} \cdot 3d^{3}
-3m^{2}n^{5} \cdot 7m^{3}n
Example 4
Multiply and write the answer in scientific notation.\left(2.7 \times 10^5\right)\left(6.04 \times 10^{13}\right)
For any base number a, and any numbers m and n as powers, a^{m} \cdot a^{n}=a^{m+n}
When multiplying terms with like bases, we add the powers.