Simplifying Variable Expressions Using Exponent Properties I
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
| [latex]{x}^{2}\cdot{x}^{3}[/latex] | |
| What does this mean? How many factors altogether? |  |
| So, we have | [latex]{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}={x}^{5}[/latex] |
| Notice that [latex]5[/latex] is the sum of the exponents, [latex]2[/latex] and [latex]3[/latex]. | [latex]{x}^{2}\cdot{x}^{3}[/latex] is [latex]{x}^{2+3}[/latex], or [latex]{x}^{5}[/latex] |
| We write: | [latex]{x}^{2}\cdot {x}^{3}[/latex] [latex-display]{x}^{2+3}[/latex-display] [latex]{x}^{5}[/latex] |
The Product Property OF Exponents
For any real number [latex]x[/latex] and any integers a and b, [latex]\left(x^{a}\right)\left(x^{b}\right) = x^{a+b}[/latex]. To multiply exponential terms with the same base, add the exponents.
Caution! When you are reading mathematical rules, it is important to pay attention to the conditions on the rule. For example, when using the product rule, you may only apply it when the terms being multiplied have the same base and the exponents are integers. Conditions on mathematical rules are often given before the rule is stated, as in this example it says "For any number x, and any integers a and b."
[latex]\begin{array}{ccc}\hfill {2}^{2}\cdot {2}^{3}& \stackrel{?}{=}& {2}^{2+3}\hfill \\ \hfill 4\cdot 8& \stackrel{?}{=}& {2}^{5}\hfill \\ \hfill 32& =& 32\hfill \end{array}[/latex]
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]
Answer: The base of both exponents is a, 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-display]\left(a^{3}\right)\left(a^{7}\right) = a^{10}[/latex-display]try it
[ohm_question]146102[/ohm_question]example
Simplify: [latex]{b}^{4}\cdot b[/latex]Answer: 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
[ohm_question]146107[/ohm_question]example
Simplify: [latex]{2}^{7}\cdot {2}^{9}[/latex]Answer: Solution
| [latex]{2}^{7}\cdot {2}^{9}[/latex] | |
| Use the product property, [latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex]. | [latex]2^{\color{red}{7+9}}[/latex] |
| Simplify. | [latex]{2}^{16}[/latex] |
try it
[ohm_question]146143[/ohm_question]example
Simplify: [latex]{y}^{17}\cdot {y}^{23}[/latex]Answer: Solution
| [latex]{y}^{17}\cdot {y}^{23}[/latex] | |
| Notice, the bases are the same, so add the exponents. | [latex]y^{\color{red}{17+23}}[/latex] |
| Simplify. | [latex]{y}^{40}[/latex] |
try it
[ohm_question]146144[/ohm_question]example
Simplify: [latex]{x}^{3}\cdot {x}^{4}\cdot {x}^{2}[/latex]Answer: 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
[ohm_question]146145[/ohm_question]
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).Simplify Expressions Using the Power Property of Exponents
We will now further expand our capabilities with exponents. We will learn what to do when a term with a power is raised to another power, and what to do when two numbers or variables are multiplied and both are raised to an exponent. We will also learn what to do when numbers or variables that are divided are raised to a power. We will begin by raising powers to powers. See if you can discover a general property. Let’s simplify [latex]\left(5^{2}\right)^{4}[/latex]. In this case, the base is [latex]5^2[/latex] and the exponent is [latex]4[/latex], so you multiply [latex]5^{2}[/latex] four times: [latex]\left(5^{2}\right)^{4}=5^{2}\cdot5^{2}\cdot5^{2}\cdot5^{2}=5^{8}[/latex] (using the Product Rule—add the exponents). [latex]\left(5^{2}\right)^{4}[/latex] is a power of a power. It is the fourth power of [latex]5[/latex] to the second power. And we saw above that the answer is [latex]5^{8}[/latex]. Notice that the new exponent is the same as the product of the original exponents: [latex]2\cdot4=8[/latex]. So, [latex]\left(5^{2}\right)^{4}=5^{2\cdot4}=5^{8}[/latex] (which equals 390,625, if you do the multiplication). Likewise, [latex]\left(x^{4}\right)^{3}=x^{4\cdot3}=x^{12}[/latex] This leads to another rule for exponents—the Power Rule for Exponents. To simplify a power of a power, you multiply the exponents, keeping the base the same. For example, [latex]\left(2^{3}\right)^{5}=2^{15}[/latex].| [latex]({x}^{2})^{3}[/latex] | |
| [latex]{x}^{2}\cdot{x}^{2}\cdot{x}^{2}[/latex] | |
| What does this mean? How many factors altogether? |  |
| So, we have | [latex]{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}={x}^{6}[/latex] |
| Notice that [latex]6[/latex] is the product of the exponents, [latex]2[/latex] and [latex]3[/latex]. | [latex]({x}^{2})^{3}[/latex] is [latex]{x}^{2\cdot3}[/latex] or [latex]{x}^{6}[/latex] |
| We write: | [latex]{\left({x}^{2}\right)}^{3}[/latex] [latex-display]{x}^{2\cdot 3}[/latex-display] [latex]{x}^{6}[/latex] |
Power Property of Exponents
If [latex]x[/latex] is a real number and [latex]a,b[/latex] are whole numbers, then [latex-display]{\left({x}^{a}\right)}^{b}={x}^{a\cdot b}[/latex-display] To raise a power to a power, multiply the exponents. Take a moment to contrast how this is different from the product rule for exponents found on the previous page.[latex]\begin{array}{ccc}\hfill {\left({5}^{2}\right)}^{3}& \stackrel{?}{=}& {5}^{2\cdot 3}\hfill \\ \hfill {\left(25\right)}^{3}& \stackrel{?}{=}& {5}^{6}\hfill \\ \hfill 15,625& =& 15,625\hfill \end{array}[/latex]
example
Simplify: 1. [latex]{\left({x}^{5}\right)}^{7}[/latex] 2. [latex]{\left({3}^{6}\right)}^{8}[/latex]Answer: 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
[ohm_question]146148[/ohm_question]Example
Simplify [latex]6\left(c^{4}\right)^{2}[/latex].Answer: Since you 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-display]6\left(c^{4\cdot 2}\right)=6c^{8}[/latex-display]Simplify Expressions Using the Product to a Power Property
We will now look at an expression containing a product that is raised to a power. Look for a pattern. Simplify this expression.[latex]\left(2a\right)^{4}=\left(2a\right)\left(2a\right)\left(2a\right)\left(2a\right)=\left(2\cdot2\cdot2\cdot2\right)\left(a\cdot{a}\cdot{a}\cdot{a}\cdot{a}\right)=\left(2^{4}\right)\left(a^{4}\right)=16a^{4}[/latex]
Notice that the exponent is applied to each factor of [latex]2a[/latex]. So, we can eliminate the middle steps.[latex]\begin{array}{l}\left(2a\right)^{4} = \left(2^{4}\right)\left(a^{4}\right)\text{, applying the }4\text{ to each factor, }2\text{ and }a\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,=16a^{4}\end{array}[/latex]
The product of two or more numbers raised to a power is equal to the product of each number raised to the same power.| [latex]{\left(2x\right)}^{3}[/latex] | |
| What does this mean? | [latex]2x\cdot 2x\cdot 2x[/latex] |
| We group the like factors together. | [latex]2\cdot 2\cdot 2\cdot x\cdot x\cdot x[/latex] |
| How many factors of [latex]2[/latex] and of [latex]x?[/latex] | [latex]{2}^{3}\cdot {x}^{3}[/latex] |
| Notice that each factor was raised to the power. | [latex]{\left(2x\right)}^{3}\text{ is }{2}^{3}\cdot {x}^{3}[/latex] |
| We write: | [latex]{\left(2x\right)}^{3}[/latex] [latex]{2}^{3}\cdot {x}^{3}[/latex] |
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-display]{\left(ab\right)}^{m}={a}^{m}{b}^{m}[/latex-display] To raise a product to a power, raise each factor to that power. How is this rule different from the power raised to a power rule? How is it different from the product rule for exponents shown above?[latex]\begin{array}{ccc}\hfill {\left(2\cdot 3\right)}^{2}& \stackrel{?}{=}& {2}^{2}\cdot {3}^{2}\hfill \\ \hfill {6}^{2}& \stackrel{?}{=}& 4\cdot 9\hfill \\ \hfill 36& =& 36\hfill \end{array}[/latex]
example
Simplify: [latex]{\left(-11x\right)}^{2}[/latex]Answer: 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
[ohm_question]146152[/ohm_question]example
Simplify: [latex]{\left(3xy\right)}^{3}[/latex]Answer: 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
[ohm_question]146154[/ohm_question]Example
Simplify. [latex]\left(2yz\right)^{6}[/latex]Answer: Apply the exponent to each number in the product. [latex-display]2^{6}y^{6}z^{6}[/latex-display]
Answer
[latex-display]\left(2yz\right)^{6}=64y^{6}z^{6}[/latex-display]Example
Simplify. [latex]\left(−7a^{4}b\right)^{2}[/latex]Answer: Apply the exponent 2 to each factor within the parentheses. [latex-display]\left(−7\right)^{2}\left(a^{4}\right)^{2}\left(b\right)^{2}[/latex-display] 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-display]\left(-7a^{4}b\right)^{2}=49a^{8}b^{2}[/latex-display]Contribute!
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Licenses & Attributions
CC licensed content, Original
- Question ID 146154, 146153, 146152. Authored by: Lumen Learning. License: CC BY: Attribution.
- Ex: Simplify Exponential Expressions Using the Power Property of Exponents. Authored by: James Sousa (mathispower4u.com). License: CC BY: Attribution.
CC licensed content, Shared previously
- Simplify Expressions Using the Product Rule of Exponents (Basic). Authored by: James Sousa (mathispower4u.com). License: CC BY: Attribution.
- Ex: Simplify Exponential Expressions Using Power Property - Products to Powers. Authored by: James Sousa (mathispower4u.com). License: CC BY: Attribution.
CC licensed content, Specific attribution
- Prealgebra. Provided by: OpenStax License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
