Learning Outcomes
- Simplify expressions using the Product Property of Exponents
- Simplify expressions using the Power Property of Exponents
- Simplify expressions using the Product to a Power Property of Exponents
Key words
- Product Property of Exponents: to multiply exponential terms with the same base, add the exponents.
- Power Property of Exponents: to raise a power to a power, multiply the exponents.
- Product to a Power Property of Exponents: to raise a product to a power, raise each factor to that power.
The Product Property of Exponents
We have already derived the properties of exponents for multiplication using integers. This property also works for variables.
For example, [latex]\color{blue}{x^4}\cdot\color{green}{x^5}=\color{blue}{x\cdot x\cdot x\cdot x}\cdot\color{green}{x\cdot x\cdot x\cdot x\cdot x}=x^9[/latex].
We restate the product property here.
The Product Property OF Exponents
For any real number [latex]x[/latex] and any rational numbers [latex]m[/latex] and [latex]n[/latex], [latex]\left(x^{m}\right)\left(x^{n}\right) = x^{m+n}[/latex].
To multiply exponential terms with the same base, add the exponents.
example
Simplify: [latex]{x}^{5}\cdot {x}^{7}[/latex]
Solution
| [latex]{x}^{5}\cdot {x}^{7}[/latex] | |
| Use the product property, [latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex]. | [latex]x^{\color{red}{5+7}}[/latex] |
| Simplify. | [latex]{x}^{12}[/latex] |
Example
Simplify.
[latex](a^{3})(a^{7})[/latex]
Solution
The base of both exponents is [latex]a[/latex], so the product rule applies.
[latex]\left(a^{3}\right)\left(a^{7}\right)[/latex]
Add the exponents with a common base.
[latex]a^{3+7}[/latex]
Answer
[latex]\left(a^{3}\right)\left(a^{7}\right) = a^{10}[/latex]
try it
example
Simplify: [latex]{b}^{4}\cdot b[/latex]
Solution
| [latex]{b}^{4}\cdot b[/latex] | |
| Rewrite, [latex]b={b}^{1}[/latex]. | [latex]{b}^{4}\cdot {b}^{1}[/latex] |
| Use the product property, [latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex]. | [latex]b^{\color{red}{4+1}}[/latex] |
| Simplify. | [latex]{b}^{5}[/latex] |
try it
try it
We can extend the Product Property of Exponents to more than two factors.
example
Simplify: [latex]{x}^{3}\cdot {x}^{4}\cdot {x}^{2}[/latex]
Solution
| [latex]{x}^{3}\cdot {x}^{4}\cdot {x}^{2}[/latex] | |
| Add the exponents, since the bases are the same. | [latex]x^{\color{red}{3+4+2}}[/latex] |
| Simplify. | [latex]{x}^{9}[/latex] |
try it
The following video shows more examples of how to use the product rule for exponents to simplify expressions.
Caution! Do not try to apply this rule to sums.
Think about the expression [latex]\left(2+3\right)^{2}[/latex]. Does [latex]\left(2+3\right)^{2}[/latex] equal [latex]2^{2}+3^{2}[/latex]?
No, it does not because of the order of operations!
[latex]\left(2+3\right)^{2}=5^{2}=25[/latex]
and
[latex]2^{2}+3^{2}=4+9=13[/latex]
Therefore, you can only use this rule when the numbers inside the parentheses are being multiplied (or divided, as we will see next).
The Power Property of Exponents
In 2.4.1, we derived the properties of exponents for multiplication. This property also works for variables.
For example, [latex]\left ( \color{blue}{x^4}\right )^{5}=\color{blue}{x^4}\color{blue}{x^4}\color{blue}{x^4}\color{blue}{x^4}\color{blue}{x^4}=x^{20}[/latex].
We restate the power property here.
Power Property of Exponents
If [latex]x[/latex] is a real number and [latex]a,b[/latex] are whole numbers, then
[latex]{\left({x}^{a}\right)}^{b}={x}^{a\cdot b}[/latex]
To raise a power to a power, multiply the exponents.
example
Simplify:
1. [latex]{\left({x}^{5}\right)}^{7}[/latex]
2. [latex]{\left({3}^{6}\right)}^{8}[/latex]
Solution
| 1. | |
| [latex]{\left({x}^{5}\right)}^{7}[/latex] | |
| Use the Power Property, [latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex]. | [latex]x^{\color{red}{5\cdot{7}}}[/latex] |
| Simplify. | [latex]{x}^{35}[/latex] |
| 2. | |
| [latex]{\left({3}^{6}\right)}^{8}[/latex] | |
| Use the Power Property, [latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex]. | [latex]3^{\color{red}{6\cdot{8}}}[/latex] |
| Simplify. | [latex]{3}^{48}[/latex] |
try it
Example
Simplify [latex]6\left(c^{4}\right)^{2}[/latex].
Solution
Since we are raising a power to a power, apply the Power Rule and multiply exponents to simplify. The coefficient remains unchanged because it is outside of the parentheses.
[latex]6\left(c^{4}\right)^{2}[/latex]
Answer
[latex]6\left(c^{4\cdot 2}\right)=6c^{8}[/latex]
Watch the following video to see more examples of how to use the power rule for exponents to simplify expressions.
The Product to a Power Property
In 2.4.1, we derived the properties of exponents for multiplication.This property also works for variables.
For example, [latex]\left ( \color{blue}{x}\color{green}{y}\right )^{4}=\color{blue}{x\cdot x\cdot x\cdot x}\cdot\color{green}{x\cdot x\cdot x\cdot x}=x^4\,y^4[/latex].
We restate the product to a power property here.
Product to a Power Property of Exponents
If [latex]a[/latex] and [latex]b[/latex] are real numbers and [latex]m[/latex] is a whole number, then
[latex]{\left(ab\right)}^{m}={a}^{m}{b}^{m}[/latex]
To raise a product to a power, raise each factor to that power.
example
Simplify: [latex]{\left(-11x\right)}^{2}[/latex]
Solution
| [latex]{\left(-11x\right)}^{2}[/latex] | |
| Use the Power of a Product Property, [latex]{\left(ab\right)}^{m}={a}^{m}{b}^{m}[/latex]. | [latex](-11)^{\color{red}{2}}x^{\color{red}{2}}[/latex] |
| Simplify. | [latex]121{x}^{2}[/latex] |
try it
example
Simplify: [latex]{\left(3xy\right)}^{3}[/latex]
Solution
| [latex]{\left(3xy\right)}^{3}[/latex] | |
| Raise each factor to the third power. | [latex]3^{\color{red}{3}}x^{\color{red}{3}}y^{\color{red}{3}}[/latex] |
| Simplify. | [latex]27{x}^{3}{y}^{3}[/latex] |
try it
If the variable has an exponent with it, use the Power Rule: multiply the exponents.
Example
Simplify. [latex]\left(−7a^{4}b\right)^{2}[/latex]
Solution
Apply the exponent 2 to each factor within the parentheses.[latex]\left(−7\right)^{2}\left(a^{4}\right)^{2}\left(b\right)^{2}[/latex]
Square the coefficient and use the Power Rule to square [latex]\left(a^{4}\right)^{2}[/latex].
[latex]49a^{4\cdot2}b^{2}[/latex]
Simplify.
[latex]49a^{8}b^{2}[/latex]
Answer
[latex]\left(-7a^{4}b\right)^{2}=49a^{8}b^{2}[/latex]
The next video shows more examples of how to simplify a product raised to a power.
Try It
Simplify.
1. [latex]\left(3x^{5}y\right)^{2}[/latex]
2. [latex]\left(-2a^{3}b\right)^{3}[/latex]
3. [latex]\left(-4y^{5}z^{3}\right)^{2}[/latex]
All of the multiplication properties of exponents can be used together and, along with the distributive property, used to simplify algebraic expressions.
Watch the following video for some examples of how to use the power and product rules of exponents to simplify and multiply expressions.
Example
Simplify: [latex]3x^3\left (4x^2-7x^5\right )[/latex]
Solution
[latex]\color{blue}{3x^3}\left (4x^2-7x^5\right )[/latex]
[latex]=\color{blue}{3x^3}\left (4x^2\right )-\color{blue}{3x^3}\left (7x^5\right )[/latex] Distribute [latex]3x[/latex] to both terms.
[latex]=3(4)\left (x^3\cdot x^2\right )-3(7)\left ( x^3\cdot x^5\right )[/latex] Rearrange using the associative and commutative properties.
[latex]=12x^{3+2}-21x^{3+5}[/latex] Multiply the constants. Add the exponents and keep the common base.
[latex]=12x^{5}-21x^{8}[/latex]
Example
Simplify: [latex]-2x^3\left (5\left (x^2\right )^4+\left ( 3x^5\right )^2\right )[/latex]
Solution
[latex]-2x^3\left (5\left (x^2\right )^4+\left ( 3x^5\right )^2\right )[/latex]
[latex]=-2x^3\left (5x^8+\left ( 3x^5\right )^2\right )[/latex]
[latex]=-2x^3\left (5x^8+9x^{10}\right )[/latex]
[latex]=-2x^3\left (5x^8 \right )-2x^3\left (9x^{10}\right )[/latex]
[latex]=-2(5)\left (x^3\cdot x^8 \right )-2(9)\left (x^3\cdot x^{10}\right )[/latex]
[latex]=-10x^{11}-18x^{13}[/latex]
Try It
Simplify: [latex]-5x^2\left (3x^4-\left ( 2x^5\right )^3\right )[/latex]
try it
try it
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