8.1: Exponential Properties Involving Products
What if you wanted to simplify a mathematical expression containing exponents, like
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CK-12 Foundation: 0801S Product of Powers
Guidance
Back in chapter 1, we briefly covered expressions involving exponents, like
Example A
Write in exponential form.
a)
b)
c)
d)
Solution
a)
b)
c)
d)
When the base is a variable, it’s convenient to leave the expression in exponential form; if we didn’t write
Let’s simplify the expressions from Example A.
Example B
Simplify.
a)
b)
c)
d)
Solution
a)
b)
c)
d)
Be careful when taking powers of negative numbers. Remember these rules:
So even powers of negative numbers are always positive. Since there are an even number of factors, we pair up the negative numbers and all the negatives cancel out.
And odd powers of negative numbers are always negative. Since there are an odd number of factors, we can still pair up negative numbers to get positive numbers, but there will always be one negative factor left over, so the answer is negative:
Use the Product of Powers Property
So what happens when we multiply one power of
So
So
You should see that when we take the product of two powers of
Product Rule for Exponents:
There are some easy mistakes you can make with this rule, however. Let’s see how to avoid them.
Example C
Multiply
Solution
Note that when you use the product rule you don’t multiply the bases. In other words, you must avoid the common error of writing
Example D
Multiply \begin{align*}2^2 \cdot 3^3\end{align*}.
Solution
\begin{align*}2^2 \cdot 3^3 = 4 \cdot 27 = 108\end{align*}
In this case, we can’t actually use the product rule at all, because it only applies to terms that have the same base. In a case like this, where the bases are different, we just have to multiply out the numbers by hand—the answer is not \begin{align*}2^5\end{align*} or \begin{align*}3^5\end{align*} or \begin{align*}6^5\end{align*} or anything simple like that.
Watch this video for help with the Examples above.
CK-12 Foundation: Products of Powers
Vocabulary
- An exponent is a power of a number that shows how many times that number is multiplied by itself. An example would be \begin{align*}2^3\end{align*}. You would multiply 2 by itself 3 times: \begin{align*}2 \times 2 \times 2\end{align*}. The number 2 is the base and the number 3 is the exponent. The value \begin{align*}2^3\end{align*} is called the power.
- Product Rule for Exponents: \begin{align*}x^n \cdot x^m = x^{(n+m)}\end{align*}
Guided Practice
Simplify the following exponents:
a. \begin{align*}(-2)^5\end{align*}
b. \begin{align*}(10x)^2\end{align*}
Solutions:
a. \begin{align*}(-2)^5=(-2)(-2)(-2)(-2)(-2)=-32\end{align*}
b. \begin{align*}(10x)^2=10^2\cdot x^2=100x^2\end{align*}
Practice
Write in exponential notation:
- \begin{align*}4 \cdot 4 \cdot 4 \cdot 4 \cdot 4\end{align*}
- \begin{align*}3x \cdot 3x \cdot 3x\end{align*}
- \begin{align*}(-2a)(-2a)(-2a)(-2a)\end{align*}
- \begin{align*}6 \cdot 6 \cdot 6 \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y\end{align*}
- \begin{align*}2 \cdot x \cdot y \cdot 2 \cdot 2 \cdot y \cdot x\end{align*}
Find each number.
- \begin{align*}5^4\end{align*}
- \begin{align*}(-2)^6\end{align*}
- \begin{align*}(0.1)^5\end{align*}
- \begin{align*}(-0.6)^3\end{align*}
- \begin{align*}(1.2)^2+5^3\end{align*}
- \begin{align*}3^2 \cdot (0.2)^3\end{align*}
Multiply and simplify:
- \begin{align*}6^3 \cdot 6^6\end{align*}
- \begin{align*}2^2 \cdot 2^4 \cdot 2^6\end{align*}
- \begin{align*}3^2 \cdot 4^3\end{align*}
- \begin{align*}x^2 \cdot x^4\end{align*}
- \begin{align*}(-2y^4)(-3y)\end{align*}
- \begin{align*}(4a^2)(-3a)(-5a^4)\end{align*}
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Base
When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression , 32 is the base, and 4 is the exponent.Exponent
Exponents are used to describe the number of times that a term is multiplied by itself.Power
The "power" refers to the value of the exponent. For example, is "three to the fourth power".Image Attributions
Here you'll learn how to write repeated multiplication in exponential form. You'll also learn how to multiply and simplify exponential expressions.