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.)

**Grade:**912

**Strand:**Number Sense and Operations

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Base
- Exponent
- Expression

### Vertical Alignment

Previous BenchmarksNextBenchmarks

### 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, $\sqrt[3]{8}$ = $\sqrt[3]{\mathrm{2\xb3}}$ 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 $\sqrt[3]{8}$ = 2^{1}which is equivalent to the equation $\sqrt[3]{8}$ = 2.

- For example, $\sqrt[3]{8}$ = $\sqrt[3]{\mathrm{2\xb3}}$ is equivalent to the equation $\sqrt[3]{8}$ = (2³)
- 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.

- For example, if evaluating (-27)

### 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 6
^{3}.

- 6
^{3 }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:

### Instructional Tasks

*Instructional Task 1 (*

*MTR.4.1*,*MTR.5.1*)

- 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?

*Instructional Task 2 (MTR.3.1,*

*MTR.4.1*)

- 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.

*Instructional Task 3 (*

*MTR.3.1*,*MTR.5.1*)

- 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 8
^{0.5}⋅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

## 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.