5.3: Laws of Exponents - Power Rules
- Page ID
- 220277
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Videos
Proof: Proof of the Power Law of Exponents
Examples: Using the Power Laws of Exponents
Example: Advanced Example of Factoring a Trinomial Within a More Complicated Expression Requiring the Laws of Exponents
Definitions and Theorems
The power rules extend the basic exponent laws to expressions in which an entire power, product, or quotient is itself raised to an exponent. Throughout, \(x\) and \(y\) denote real numbers and \(m\) and \(n\) denote integers.
For a real number \(x\) and integers \(m\) and \(n\),\[\left(x^{m}\right)^{n} = x^{mn}.\nonumber\]That is, a power raised to a power is found by multiplying the exponents.
For real numbers \(x\) and \(y\) and an integer \(n\),\[\left(xy\right)^{n} = x^{n}y^{n}.\nonumber\]That is, an exponent distributes across each factor of a product.
For real numbers \(x\) and \(y\) with \(y \neq 0\), and an integer \(n\),\[\left(\dfrac{x}{y}\right)^{n} = \dfrac{x^{n}}{y^{n}}.\nonumber\]That is, an exponent distributes across the numerator and the denominator of a quotient.
When a product inside parentheses includes a numerical coefficient, the exponent applies to that coefficient as well. For example, \(\left(3x\right)^{2} = 3^{2}x^{2} = 9x^{2}\), not \(3x^{2}\). Forgetting to raise the coefficient is one of the most frequent errors with the power of a product rule.
The power of a product and power of a quotient rules apply only to multiplication and division. An exponent does not distribute across a sum or difference. In general,\[\left(x+y\right)^{n} \neq x^{n} + y^{n}.\nonumber\]For instance, \(\left(2+3\right)^{2} = 5^{2} = 25\), whereas \(2^{2} + 3^{2} = 4 + 9 = 13\).
Examples
Simplify \(\left(x^{4}\right)^{3}\).
- Answer
- By the power of a power rule, multiply the exponents:\[\left(x^{4}\right)^{3} = x^{4 \cdot 3} = x^{12}.\nonumber\]
Simplify \(\left(3xy^{2}\right)^{3}\).
- Answer
- Distribute the exponent to each factor by the power of a product rule, then apply the power of a power rule to \(y^{2}\):\[\left(3xy^{2}\right)^{3} = 3^{3}\,x^{3}\left(y^{2}\right)^{3} = 27x^{3}y^{6}.\nonumber\]The coefficient \(3\) is raised to the third power as well, giving \(3^{3} = 27\).
Simplify \(\left(\dfrac{x}{y}\right)^{5}\), where \(y \neq 0\).
- Answer
- Apply the power of a quotient rule, distributing the exponent to the numerator and the denominator:\[\left(\dfrac{x}{y}\right)^{5} = \dfrac{x^{5}}{y^{5}}.\nonumber\]
Simplify \(\left(2x^{3}y\right)^{4}\).
- Answer
- Distribute the exponent to each factor, then multiply exponents where a power is raised to a power:\[\left(2x^{3}y\right)^{4} = 2^{4}\left(x^{3}\right)^{4}y^{4} = 16x^{12}y^{4}.\nonumber\]Note that \(x^{3}\) becomes \(x^{3 \cdot 4} = x^{12}\), and the factor \(y = y^{1}\) becomes \(y^{4}\).
Simplify \(\left(\left(x^{2}\right)^{3}\right)^{2}\).
- Answer
- Apply the power of a power rule repeatedly, multiplying all three exponents together:\[\left(\left(x^{2}\right)^{3}\right)^{2} = x^{2 \cdot 3 \cdot 2} = x^{12}.\nonumber\]
Simplify \(\left(\dfrac{2x^{4}}{3y^{2}}\right)^{3}\), where \(y \neq 0\).
- Answer
- Distribute the exponent to every factor in the numerator and denominator, then simplify each power:\[\left(\dfrac{2x^{4}}{3y^{2}}\right)^{3} = \dfrac{2^{3}\left(x^{4}\right)^{3}}{3^{3}\left(y^{2}\right)^{3}} = \dfrac{8x^{12}}{27y^{6}}.\nonumber\]Both coefficients are cubed: \(2^{3} = 8\) and \(3^{3} = 27\).
MyOpenMath Practice
Sources
Several parts of this text use modifications from the following source:
- Wikipedia article: "Exponentiation"
This source is released under the Creative Commons Attribution-Share-Alike License 4.0.