Curriculum / Math / 9th Grade / Unit 6: Exponents and Exponential Functions / Lesson 6
Math
Unit 6
9th Grade
Lesson 6 of 22
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Lesson Notes
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Define rational exponents and convert between rational exponents and roots.
The core standards covered in this lesson
N.RN.A.1 — Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5<sup>1/3</sup> to be the cube root of 5 because we want (5<sup>1/3</sup>)³ = 5(<sup>1/3</sup>)³ to hold, so (5<sup>1/3</sup>)³ must equal 5.
N.RN.A.2 — Rewrite expressions involving radicals and rational exponents using the properties of exponents.
The foundational standards covered in this lesson
8.EE.A.1 — Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² × 3<sup>-5</sup> = 3<sup>-3</sup> = 1/3³ = 1/27.
8.EE.A.2 — Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
8.NS.A.1 — Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
25-30 minutes
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..
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. $${\sqrt{(3x)^5}}$$
A set of suggested resources or problem types that teachers can turn into a problem set
15-20 minutes
Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
5-10 minutes
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 $$8$$. "
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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.
The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.
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Write equivalent radical and rational exponent expressions. Identify quantities as rational or irrational.
Topic A: Exponent Rules, Expressions, and Radicals
Use exponent rules to analyze and rewrite expressions with non-negative exponents.
Standards
8.EE.A.1
Add and subtract polynomial expressions using properties of operations.
A.APR.A.1
Multiply polynomials using properties of exponents and properties of operations.
Solve mathematical applications of exponential expressions.
Use negative exponent rules to analyze and rewrite exponential expressions.
8.EE.A.1A.SSE.A.2
N.RN.A.1N.RN.A.2
N.RN.B.3
Simplify radical expressions.
N.RN.A.2
Multiply and divide rational exponent expressions and radical expressions.
N.RN.A.2N.RN.B.3
Add and subtract rational exponent expressions and radical expressions.
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Topic B: Arithmetic and Geometric Sequences
Describe and analyze sequences given their recursive formulas.
F.BF.A.2F.IF.A.2F.IF.A.3
Write recursive formulas for sequences, including the Fibonacci sequence.
Define arithmetic and geometric sequences, and identify common ratios and common differences in sequences.
F.BF.A.2F.LE.A.2
Write explicit rules for arithmetic sequences and translate between explicit and recursive formulas.
F.BF.A.2F.IF.A.3F.LE.A.2
Write explicit rules for geometric sequences and translate between explicit and recursive formulas.
Topic C: Exponential Growth and Decay
Compare rates of change in linear and exponential functions shown as equations, graphs, and situations.
A.SSE.A.1F.IF.C.9F.LE.A.1F.LE.A.3
Write linear and exponential models for real-world and mathematical problems.
A.SSE.A.1F.LE.A.1F.LE.A.2F.LE.B.5
Graph exponential growth functions and write exponential growth functions from graphs.
F.BF.B.3F.IF.C.7.E
Write exponential growth functions to model financial applications, including compound interest.
F.IF.C.8.BF.LE.A.2F.LE.B.5
Write, graph, and evaluate exponential decay functions.
F.BF.B.3F.IF.C.7.EF.IF.C.8.BF.LE.A.1.C
Identify features of exponential decay in real-world problems.
F.IF.C.8.BF.LE.A.1.C
Solve exponential growth and exponential decay application problems.
F.IF.C.8.BF.LE.A.1F.LE.A.2
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