Lesson 5 | Exponents and Scientific Notation | 8th Grade Mathematics

Objective

Apply the product of powers rule and the quotient of powers rule to write equivalent, simplified exponential expressions.

Common Core Standards

Core Standards

The core 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^-5 = 3^-3 = 1/3³ = 1/27.

Expressions and Equations

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.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Investigate, determine, and apply the general rule for product of powers: $${x^mx^n=x^{m+n}}$$.
Investigate, determine, and apply the general rule for quotient of powers: $${{x^m\over{x^n}}=x^{m-n}}$$

Tips for Teachers

Suggestions for teachers to help them teach this lesson

These Anchor Problems can be used in a variety of ways, but one effective approach is to have students lead the work rather than the teacher. Similar to the previous lesson, students can investigate and explore these problems, but now with the added objective of determining an approach to solving problems in general.
Once students have experimented with the problems and found a generalization, then provide them with the name of the rule and the general form.
Throughout the next few lessons, the problems will become increasingly complex as new rules are learned and named. It is important for students to be able to look at an exponential expression and see the structure in the problem that can help them understand what to do (MP.7). This is more important and will be more useful to students than memorizing the rules. You may consider having students create posters or other visual displays that they can refer to throughout the unit.
Note that negative exponents are not included in this lesson, as they will be introduced in Lesson 8.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Write an equivalent, simplified form of each of the following:

a. $${6^3\times6^5}$$

b. $${\left (-{1\over4} \right )^4\left (-{1\over4} \right )^3}$$

c. $${12^a\cdot 12^b}$$

d. $${x^mx^n}$$

Guiding Questions

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Problem 2

Write an equivalent, simplified form of each of the following:

a. $${5^3\cdot 25\cdot 5^4}$$

b. $${5^63^2\times3\times5^3}$$

c. $${(2xy)(x^2y^3)(2y^4)}$$

Guiding Questions

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Problem 3

Write an equivalent, simplified form of each of the following:

a. $${8^6\over8^3}$$

b. $${{\left (-{1\over2} \right )^9}\over{\left (-{1\over2} \right )^5}}$$

c. $${\frac{6^{4}5^{7}}{6\cdot 2^{5}}}$$

d. $${x^m\over x^n}$$, where ($$m > n$$)

Guiding Questions

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Problem 4

Your teacher writes the following two expressions on the board:

Your teacher asks the class if these two expressions are equivalent. Half of the class says yes, and half says no. What do you think? Convince the other half of the class that you are correct.

Guiding Questions

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References

Nathan Kraft's Blog Exponent Rules are Lame

Exponent Rules are Lame is made available on Nathan Kraft's Blog. Accessed Aug. 4, 2017, 10:11 a.m..

Problem Set

A set of suggested resources or problem types that teachers can turn into a problem set

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Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Simplify the following expressions.

a. $$17^x(17^y)$$

b. $$2^4(32)$$

c. $$5^a\over 5^b$$

d. $$3^6÷27$$

Student Response

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Additional Practice

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.

Include problems where students are given an exponential expression and must write a more complex, equivalent expression (for example, write an expression equivalent to $${3^{2+4},x^{5-2},y^8}$$).

EngageNY Mathematics Grade 8 Mathematics > Module 1 > Topic A > Lesson 2 — Exercises 1-32, Problem Set 3-5
Kuta Software Free Pre-Algebra Worksheets Exponents and Radicals — Multiplication property of exponents
Open Middle Properties of Integer Exponents 2
EngageNY Mathematics Grade 8 Mathematics > Module 1 > Topic A > Lesson 2 — Optional Fluency Exercises
Open Middle Properties of Integer Exponents
Kuta Software Free Pre-Algebra Worksheets Exponents and Radicals — Division property of exponents (Some of these include quotients resulting in a negative difference. These can be adjusted to only include positive exponents, or have students simplify by expanding and leaving the remaining bases in the denominator.)

Lesson 4

Lesson 6

Lesson Map

Topic A: Review of Exponents

Review exponent notation and identify equivalent exponential expressions.

8.EE.A.1

Evaluate numerical and algebraic expressions with exponents using the order of operations.

8.EE.A.1

Investigate patterns of exponents with positive/negative bases and even/odd bases.

8.EE.A.1

Topic B: Properties of Exponents

Investigate exponent patterns to write equivalent expressions.

8.EE.A.1

Apply the product of powers rule and the quotient of powers rule to write equivalent, simplified exponential expressions.

8.EE.A.1

Apply the power of powers rule and power of product rule to write equivalent, simplified exponential expressions.

8.EE.A.1

Reason with zero exponents to write equivalent, simplified exponential expressions.

8.EE.A.1

Reason with negative exponents to write equivalent, simplified exponential expressions.

8.EE.A.1

Simplify and write equivalent exponential expressions using all exponent rules.

8.EE.A.1

Topic C: Scientific Notation

Write large and small numbers as powers of 10.

8.EE.A.3 8.EE.A.4

Define and write numbers in scientific notation.

8.EE.A.3

Compare numbers written in scientific notation.

8.EE.A.3 8.EE.A.4

Multiply and divide with numbers in scientific notation. Interpret scientific notation on calculators.

8.EE.A.4

Add and subtract with numbers in scientific notation.

8.EE.A.4

Solve multi-step applications using scientific notation and properties of exponents.

8.EE.A.1 8.EE.A.3 8.EE.A.4

Exponents and Scientific Notation

Objective

Common Core Standards

Core Standards

Expressions and Equations

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Problem 4

Guiding Questions

References

Problem Set

Target Task

Student Response

Additional Practice

Lesson Map

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