Define rational exponents and convert between rational exponents and roots.
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Below is an equation that is not true.
$${{{{{10}0}^{1\over2}}}=50}$$
a. Why is the statement incorrect? What do you think the correct value of $${{{{10}0}^{1\over2}}}$$ is?
b. Consider the following pattern. Where does $${{{{10}0}^{1\over2}}}$$ fit in?
$${{{10}0}^3=1,000,000}$$
$${{{10}0}^2={10},000}$$
$${{{10}0}^1={{10}0}}$$
$${{{10}0}^0=1}$$
c. Consider rewriting the base $${{10}0}$$ as a power of $${10}$$. How does this shed light on the value of $${{{{10}0}^{1\over2}}}$$?
$${{{{10}0}^{1\over2}}}=(\square)^{1\over2}$$
d. Try out these other rational exponents:
$${25^{1\over2}}$$ $${144^{1\over2}}$$ $${8^{1\over3}}$$
Mistakes to the Half Power is made available by Andrew Stadel on Divisible by 3 under the CC BY-NC-SA 3.0 license. Accessed May 17, 2018, 10:54 a.m..
Modified by Fishtank Learning, Inc.All of the following equations are true.
$${\sqrt{x}=x^{1\over2}}$$ $${\sqrt[3]{x}=x^{1\over3}}$$ $${(\sqrt{x})^2=x}$$ $${x^{2\over3}=\sqrt[3]{x^2}}$$
Determine a general statement to represent the relationship between a radical and its exponential expression.
Write the radicals in exponential form and write the exponentials in radical form.
a. $${5^{6\over5}}$$
b. $${4^{-{2\over3}}}$$
c. $${2n^{2\over5}}$$
d. $${\sqrt[3]{7^2}}$$
e. $${1\over{\sqrt[3]{5}}}$$
f. $${\sqrt{(3x)^5}}$$
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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.
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Henry explains why $${4^{3\over2}=8}$$:
"I know that $${4^3}$$ is $${{64}}$$ and the square root of $${{64}}$$ is $$8$$."
Here is Henrietta’s explanation for why $${4^{3\over2}=8}$$:
"I know that $${\sqrt4=2}$$ and the cube of $$2$$ is $$8$$. "
Evaluating Exponential Expressions, accessed on May 18, 2018, 12:33 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.