5.05 Rational exponents
Ideas
Rational exponents
In previous lessons, we have focused on integer exponents, but exponents can also be fractions. We call these rational exponents.
Exploration
Consider the following statements:⬚\cdot 2=1
\left(\sqrt{3}\right)^2 =3^1=\left(3^{\text{⬚}}\right)^2
\left(\sqrt[3]{8}\right)^3=8^1=\left(8^{\text{⬚}}\right)^3
- Which exponent law is being used?
- What values would make each statement true?
- Generalize the relationships by completing these statements: \sqrt{a}=a^{\frac{⬚}{⬚}} and \sqrt[3]{a}=a^{\frac{⬚}{⬚}}
Let's formalize how to work with rational exponents of \frac{1}{2} and \frac{1}{3}.
First, using the product law, we have:2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}}=2^{\frac{1}{2}+\frac{1}{2}} = 2^1=2
The definition of a square root is a value which multiplies by itself to give the original number. We know that\sqrt{2} \cdot \sqrt{2} = \left(\sqrt{2}\right)^2=2^1=2
Comparing these two statements, we see that2^{\frac{1}{2}} = \sqrt{2}
We can confirm this by using the power rule:\left(2^{\frac{1}{2}}\right)^2 = 2^{\frac{1}{2} \cdot 2} = 2^1 = 2
In a similar way, we can look at the value 2^{\frac{1}{3}}. Using the product law:2^{\frac{1}{3}} \cdot 2^{\frac{1}{3}} \cdot 2^{\frac{1}{3}} =2^{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}}= 2
From the definition of a cube root:\sqrt[3]{2} \cdot \sqrt[3]{2} \cdot \sqrt[3]{2} =\left(\sqrt[3]{2}\right)^3= 2
Once again, by comparison we see that 2^{\frac{1}{3}} = \sqrt[3]{2}
These results can be generalized to:a^{\frac{1}{2}} = \sqrt{a} and a^{\frac{1}{3}} = \sqrt[3]{a}
Notice that the index of the radical becomes the denominator of the rational exponent. When there is no index shown, it is a square root.
We can use these rules for rewriting radicals along with the laws of exponents to simplify expressions involving radicals and rational exponents.
| \text{Product rule} | a^{m} \cdot a^{n} = a^{m+n} |
| \text{Quotient rule} | \dfrac{{a}^{m}}{{a}^{n}}=a^{m-n} |
| \text{Power rule} | \left(a^{m}\right)^{n} = a^{mn} |
| \text{Power of a product} | \left(a b\right)^{m} = a^{m} \cdot b^{m} |
| \text{Power of a quotient} | \left(\dfrac{a} {b}\right)^{m} =\dfrac {a^{m}} {b^{m}} |
| \text{Identity exponent} | a^1=a |
| \text{Zero rule} | a^0=1 |
| \text{Negative exponent rule} | a^{- {m}}=\dfrac{1}{a^{m}} |
| \text{Rational exponent} | a^{\frac{1}{n}}=\sqrt[n]{a} |
Examples
Example 1
For 36^\frac{1}{2}
Rewrite in radical form.
Evaluate 36^\frac{1}{2}
Example 2
Write \sqrt[3]{6} in exponential form.
Example 3
Evaluate without a calculator.
\left(-216\right)^\frac{1}{3}
8^\frac{5}{3}
\left(27^\frac{1}{3}\right)^2\cdot \dfrac{27^\frac{1}{2}}{27^\frac{5}{6}}
In addition to the laws of exponents we have seen, we can simplify rational exponents by converting to and from radical form.a^{\frac{1}{2}} = \sqrt{a} and a^{\frac{1}{3}} = \sqrt[3]{a}
We can simplify expressions involving rational exponents using the laws of exponents.