# MA.912.NSO.1.1

Extend previous understanding of the Laws of Exponents to include rational exponents. Apply the Laws of Exponents to evaluate numerical expressions and generate equivalent numerical expressions involving rational exponents.

### Clarifications

Clarification 1: Instruction includes the use of technology when appropriate.

Clarification 2: Refer to the K-12 Formulas (Appendix E) for the Laws of Exponents.

Clarification 3: Instruction includes converting between expressions involving rational exponents and expressions involving radicals.

Clarification 4:Within the Mathematics for Data and Financial Literacy course, it is not the expectation to generate equivalent numerical expressions.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Number Sense and Operations
Status: State Board Approved

## Benchmark Instructional Guide

• Base
• Exponent
• Expression

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students generated equivalent numerical expressions and evaluated expressions using the Laws of Exponents with integer exponents. In Algebra I, students work with rational-number exponents. In later courses, students extend the Laws of Exponents to properties of logarithms.
• Instruction includes using the terms Laws of Exponents and properties of exponents interchangeably.
• Instruction includes student discovery of the patterns and the connection to mathematical operations and the inverse relationship between powers and radicals (MTR.5.1)
• Problem types include having a fraction, integer or whole number as an exponent.
• Students should make the connection of the root being equivalent to  unit fraction exponent (MTR.4.1).
• For example, $\sqrt[3]{8}$ = $\sqrt[3]{2³}$ is equivalent to the equation $\sqrt[3]{8}$ = (2³)$\frac{\text{1}}{\text{3}}$ which is equivalent to the equation $\sqrt[3]{8}$ = 2$\frac{\text{3}}{\text{1}}$.$\frac{\text{1}}{\text{3}}$ which is equivalent to the equation $\sqrt[3]{8}$ = 21 which is equivalent to the equation $\sqrt[3]{8}$ = 2.

• When evaluating, students should be encouraged to approach from different entry points and discuss how they are different but equivalent strategies (MTR.2.1).

• For example, if evaluating (-27)$\frac{\text{2}}{\text{3}}$  students can either take the cube root of -27 first or raise -27 to the second power first.

### Common Misconceptions or Errors

• Students may not understand the difference between an expression and an equation.
• Students may try to perform operations on bases as well as exponents.
• Students may multiply the base by the exponent instead of understanding that the exponent is the number of times the base occurs as a factor.
• Students may not truly understand exponents that are zero or negative.

### Strategies to Support Tiered Instruction

• Teacher provides a review of the relationship between the base and the exponent by modeling an example of operations using a base and exponent.
• For example, determine the numerical value of 63

• 63 which is equivalent to 6 6 6 which is equivalent to 216.
• Teacher provides exploration of the rules of exponents through patterns. A strategy for developing meaning for integer exponents by making use of patterns is shown below:

• Teacher provides exploration of the rules of rational exponents through patterns. A strategy for developing meaning for rational exponents by making use of patterns is shown below:

• Part A. Think about when solving an equation with a radical. What is the inverse operation of a square root? Of a cube root?
• Part B. Given the expression $\sqrt[3]{27}$, express 27 as a prime number with natural-number exponent.
• Part C. How can we use the information from Part A and B to convert $\sqrt[3]{27}$ to exponential form?
• Part A. Evaluate 64$\frac{\text{1}}{\text{3}}$ by first writing 64 as a power of 2 and using the properties of exponents.
• Part B. Evaluate 64$\frac{\text{1}}{\text{3}}$ using a calculator.
• Part C. Explain your process in both Part A and Part B. Define powers with fractional exponents in your own words.

• Part A. Given $f$($x$) = 32$x$, evaluate $f$(0), $f$(0.2), $f$(0.4), $f$(0.8) and $f$(1) without the use of a calculator.
• Part B. Graph the function $f$ in the domain 0 ≤ $x$ ≤ 1.
• Part C. Between which two values in Part A would $f$(0.5) be? Which one would it be closer to on the graph and why?

### Instructional Items

Instructional Item 1
• Evaluate the numerical expression (64)$\frac{\text{4}}{\text{3}}$
Instructional Item 2
• Rewrite 80.5  2$\frac{\text{2}}{\text{5}}$  as a single power of 2.
Instructional Item 3
• Choose all of the expressions that are equivalent to 7$\frac{\text{5}}{\text{12}}$
• a. (49$\frac{\text{1}}{\text{3}}$)(7$\frac{\text{1}}{\text{4}}$)
• b. (7$\frac{\text{2}}{\text{3}}$)(7$\frac{\text{1}}{\text{4}}$
• c. 7(7$\frac{\text{1}}{\text{4}}$)
• d.
• e.

Instructional Item 4

•  Evaluate the numerical expression (−$\frac{\text{729}}{\text{64}}$)$\frac{\text{2}}{\text{3}}$.

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*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1200310: Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200320: Algebra 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200380: Algebra 1-B (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912090: Access Algebra 1B (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
1200315: Algebra 1 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200385: Algebra 1-B for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912075: Access Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
1200388: Mathematics for Data and Financial Literacy Honors (Specifically in versions: 2022 and beyond (current))
1200384: Mathematics for Data and Financial Literacy (Specifically in versions: 2022 and beyond (current))
7912120: Access Mathematics for Data and Financial Literacy (Specifically in versions: 2022 - 2023, 2023 and beyond (current))
1200710: Mathematics for College Algebra (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.NSO.1.AP.1: Evaluate numerical expressions involving rational exponents.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

Roots and Exponents:

Students are asked to rewrite the square root of five in exponential form and justify their choice of exponent.

Type: Formative Assessment

Rational Exponents and Roots:

Students asked to show that two forms of an expression (exponential and radical) are equivalent.

Type: Formative Assessment

## Lesson Plan

This lesson unit is intended to help you assess how well students are able to:

• Use the properties of exponents, including rational exponents, and manipulate algebraic statements involving radicals.
• Discriminate between equations and identities.

There is also an opportunity to consider the role of the imaginary number , but this is optional.

Type: Lesson Plan

## MFAS Formative Assessments

Rational Exponents and Roots:

Students asked to show that two forms of an expression (exponential and radical) are equivalent.

Roots and Exponents:

Students are asked to rewrite the square root of five in exponential form and justify their choice of exponent.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.