Examples
The expression is equivalent to the expression which is equivalent to .Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K12 Glossary
 Base
 Expression
 Exponent
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In grade 8, students generated equivalent algebraic expressions using the Laws of Exponents with integer exponents. In Algebra I, students expand this work to include rationalnumber exponents. In later courses, students extend the Laws of Exponents to algebraic expressions with 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 (MTR.5.1).
 Students should be able to fluently apply the Laws of Exponents in both directions.
 For example, students should recognize that $a$^{6 }is the quantity ($a$^{3})^{2}; this may helpful when students are factoring a difference of squares.
 When generating equivalent expressions, students should be encouraged to approach from different entry points and discuss how they are different but equivalent strategies (MTR.2.1).
 The expectation for this benchmark does not include the conversion of an algebraic expression from exponential form to radical form and from radical form to exponential form.
Common Misconceptions or Errors
 Students may not understand the difference between an expression and an equation.
 Students may not have fully mastered the Laws of Exponents and understand the mathematical connections between the bases and the exponents.
 Student may believe that with the introduction of variables, the properties of exponents differ from numerical expressions.
Strategies to Support Tiered Instruction
 Instruction includes the opportunity to distinguish between an expression and an
equation. These should be captured in a math journal.
 For example, when generating equivalent expressions, place an equal sign in between the expressions and label each expression and the equation.
 Instruction provides opportunities to write each term in expanded form first and then use
Laws of Exponents to combine like factors. It may also be helpful to chunk each step.
 For example, to rewrite the expression (8$x$^{3})^{2 }with one exponent, write out (8)($x$)($x$)($x$)(8)($x$)($x$)($x$) and then use the commutative property to write (8)(8)($x$)($x$)($x$)($x$)($x$)($x$) = 64$x$^{6}.
 Teacher provides instruction for problems that require multiple applications of the Laws of Exponents by chunking the steps so that students are applying one property at each time and explaining the property applied. Each time ask students to identify the property of exponents that they applied.
 Teacher provides students sidebyside problems, one with variable bases and the other
choosing a value for the variable. As students work through the problems, ask them about
the similarities in the problemsolving process.
 For example, teacher can model generating equivalent expressions like the ones below.

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}.
 For example, determine the numerical value of 6^{3}.
 6^{3} which is equivalent to 6 ⋅ 6 ⋅ 6 which is equivalent to 216.
Instructional Tasks
 Given the function $h$($x$) = 10^{0.2$x$ }, what is the rate of growth or decay?
 Part B. Compare your method of simplifying with a partner.
Instructional Items
 Instructional Item 1
 Given the algebraic expression 2.3^{2$t$−1}, create an equivalent expression.
 Instructional Item 2
 Use the properties of exponents to create an equivalent expression for the given expression shown below with no variables in the denominator.
(64$x$²)^{−$\frac{\text{1}}{\text{6}}$}(32$x$^{5})^{−$\frac{\text{2}}{\text{5}}$}
*The strategies, tasks and items included in the B1GM are examples and should not be considered comprehensive.
Related Courses
Related Access Points
Related Resources
Formative Assessments
ProblemSolving Task
MFAS Formative Assessments
Students are asked to transform an exponential expression so that the rate of change corresponds to a different time interval.
Students are asked to use the properties of exponents to show that two expressions are equivalent and compare the two functions in terms of what each reveals.
Students are asked to convert numerical expressions from radical to exponential form.
Students are asked to convert numerical expressions from exponential to radical form.
Students are asked to convert a product of a radical and exponential expression to a single power of two.
Students are asked to rewrite expressions involving radicals and rational exponents in equivalent forms.
Students asked to show that two forms of an expression (exponential and radical) are equivalent.
Students are asked to rewrite the square root of five in exponential form and justify their choice of exponent.
Student Resources
ProblemSolving Task
The purpose of this task is to identify the structure in the two algebraic expressions by interpreting them in terms of a geometric context. Students will have likely seen this type of process before, so the principal source of challenge in this task is to encourage a multitude and variety of approaches, both in terms of the geometric argument and in terms of the algebraic manipulation.
Type: ProblemSolving Task
Parent Resources
ProblemSolving Task
The purpose of this task is to identify the structure in the two algebraic expressions by interpreting them in terms of a geometric context. Students will have likely seen this type of process before, so the principal source of challenge in this task is to encourage a multitude and variety of approaches, both in terms of the geometric argument and in terms of the algebraic manipulation.
Type: ProblemSolving Task