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:
$${25^{1\over2}}$$ $${144^{1\over2}}$$ $${8^{1\over3}}$$
This problem is meant to engage students in conversation and experimentation around how to work with rational exponents. Give students some time with part (a) on their own or in small groups. You can then offer the strategies in parts (b) and (c) to move the discussion along if students have not discovered those strategies on their own. There is a good example of how this conversation happened in the blog author’s classroom in the link below.
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, 2:54 p.m..
Modified by The Match Foundation, 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. They can be used to create a problem set for class (for non-Fishtank Plus users), or as supplementary or additional resources to the pre-made Problem Set (for Fishtank Plus users).
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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, 4:33 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.