Dr. Sarah Ledford Mathematics Educator Sarah@SarahLikesMath.com Exponent Rules Dr. Sarah Ledford Mathematics Educator Sarah@SarahLikesMath.com

Prior Standards In 5th grade, students use whole number exponents to denote powers of 10. They also write numbers in expanded form such 345.678 as (3 x 100) + (4 x 10) + (5 x 1) + (6 x 1/10) + (7 x 1/100) + (8 x 1/1000). In 6th grade, students work with numerical expressions involving whole number exponents.

82 = 8 x 8 = 64 In the calculator: 8 ^ 2 Exponents 82 = 8 x 8 = 64 In the calculator: 8 ^ 2

Exponents 32 = 3 x 3 23 = 2 x 2 x 2 34 = 3 x 3 x 3 x 3 43 = 4 x 4 x 4 56 = 5 x 5 x 5 x 5 x 5 x 5 63 = 6 x 6 x 6 In your own words? In your own words? The power/exponent tells you how many of the base that you multiply together.

Exponents

The Calculator is always Correct? My class was asked to simplify the expression 5 + 42 – 6 ÷ 3. James’ calculator showed the answer to be 10. Alexis’ calculator showed 19. Lucy didn’t use a calculator but got 5. Jorge also didn’t use a calculator and got 25. Who is correct? From GADOE 6th grade unit 3

The Calculator is always Correct? The correct solution is 19. This problem shows the importance of having an order of operations because without them, we can get several different answers. A scientific calculator can apply the order of operations where a regular four-function calculator can not.

Equivalent? Determine which of the two expressions are equivalent: 22  32 – 23 – 1 22  (32 – 23)– 1 (2  3)2 – 23 – 1 From GADOE 6th grade unit 3 The first expression is equal to 27, the second expression is equal to 3, the third expression is equal to 27, so the second expression is not equivalent to either of the other two expressions (but the first and third expressions are equivalent.)

Exponent Experimentation 2 https://www.illustrativemathematics.org/content- standards/6/EE/A/1/tasks/2224 Here are some different ways to write the value 16: 24 12 – (21 + 22) + 500 ÷ 50 2/3 x 481 – (1 + 3)2 23 + 23

Exponent Experimentation 2 Find at least three different ways to write each value below. Include at least one exponent in all of the expressions you write. 81 25 64/9 You can have your students write their ways on the board or on chart paper. Then, ask them to check to make sure that all the ways written are correct. If any are incorrect, figure out how to correct them. It is an open-ended question & your students come up with the “problems” for everyone to simplify!!

Exponent of 1 You can use your calculator to compute these if needed: 21 = ?? 31 = ?? 51 = ?? 81 = ?? a1 = ?? If the exponent tells us how many of the base we have to multiply, then an exponent of 1 tells us we have only 1 of the base number. In your own words? Anything to the power of 1 is itself.

Product Rule What does each statement mean? 23 x 24 = ? (2 x 2 x 2) x 24 = ? (2 x 2 x 2) x (2 x 2 x 2 x 2) = ? Are the parentheses necessary? 2 x 2 x 2 x 2 x 2 x 2 x 2 = ? How many 2’s are we multiplying together? 27 = ?

Product Rule What does each statement mean? 32 x 34 = ? (3 x 3) x 34 = ? (3 x 3) x (3 x 3 x 3 x 3) = ? Are the parentheses necessary? 3 x 3 x 3 x 3 x 3 x 3= ? How many 3’s are we multiplying together? 36 = ?

Product Rule What does each statement mean? 41 x 42 = ? 4 x 42 = ? 4 x (4 x 4) = ? Are the parentheses necessary? 4 x 4 x 4 = ? How many 4’s are we multiplying together? 43 = ?

Product Rule 23 x 24 = 27 32 x 34 = 36 41 x 42 = 43 Observations? Can you come up with a rule that will always work? am x an = a(m + n) In your own words? When multiplying expressions with the same base, you can add the exponents.

What does each statement mean? Quotient Rule What does each statement mean?

What does each statement mean? 2 x 2 x 2 x 2 = ? 24 = ? Quotient Rule What does each statement mean? 2 x 2 x 2 x 2 = ? 24 = ?

What does each statement mean? Quotient Rule What does each statement mean?

What does each statement mean? 4 x 4 = ? 42 = ? Quotient Rule What does each statement mean? 4 x 4 = ? 42 = ?

What does each statement mean? Quotient Rule What does each statement mean?

What does each statement mean? 3 = ? 31 = ? Quotient Rule What does each statement mean? 3 = ? 31 = ?

Observations? Can you come up with a rule that will always work? Quotient Rule Observations? Can you come up with a rule that will always work?

Quotient Rule Observations? Can you come up with a rule that will always work? In your own words? When dividing expressions with the same base, subtract the exponents.

Power Rule What does each statement mean? (23)4 = ? (2 x 2 x 2)4 = ? (2 x 2 x 2) x (2 x 2 x 2) x (2 x 2 x 2) x (2 x 2 x 2) = ? Are the parentheses necessary? 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = ? How many 2’s are we multiplying together? 212 = ?

Power Rule What does each statement mean? (32)3 = ? (3 x 3)3 = ? (3 x 3) x (3 x 3) x (3 x 3) = ? Are the parentheses necessary? 3 x 3 x 3 x 3 x 3 x 3 = ? How many 3’s are we multiplying together? 36 = ?

Power Rule What does each statement mean? (54)2 = ? (5 x 5 x 5 x 5)2 = ? (5 x 5 x 5 x 5) x (5 x 5 x 5 x 5) = ? Are the parentheses necessary? 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 = ? How many 5’s are we multiplying together? 58 = ?

Can you come up with a rule that will always work? Power Rule (23)4 = 212 (32)3 = 36 (54)2 = 58 Observations? Can you come up with a rule that will always work? (am)n = a(m x n) In your own words? When raising a power to another power, you can multiply the powers together.

Expanded Power Rule What does each statement mean? (3z)2 = ? 3z x 3z = ? (3 x z) x (3 x z) = ? (3 x 3) x (z x z) = ? 32 x z2 = ? 32 z2 = ? 9 z2

Expanded Power Rule What does each statement mean? (2de)3 = ? 2de x 2de x 2de = ? (2 x d x e) x (2 x d x e) x (2 x d x e) = ? (2 x 2 x 2) x (d x d x d) x (e x e x e) = ? 23 x d3 x e3 = ? 23 d3 e3 = ? 8 d3 e3

What does each statement mean? Expanded Power Rule What does each statement mean?

What does each statement mean? Expanded Power Rule What does each statement mean?

What does each statement mean? Expanded Power Rule What does each statement mean?

What does each statement mean? Expanded Power Rule What does each statement mean?

What does each statement mean? Expanded Power Rule What does each statement mean?

Can you come up with rules that always work? Expanded Power Rule (3z)2 = 32 z2 (2de)3 = 23 d3 e3 Observations? Can you come up with rules that always work? (ab)m = am bm

Can you come up with rules that always work? Expanded Power Rule (3z)2 = 32 z2 (2de)3 = 23 d3 e3 Observations? Can you come up with rules that always work? (ab)m = am bm In your own words? When the base is in parentheses and there are more than value making up the base, the exponent goes to all of those values in the base.

Exploring Powers of 10 Adapted from GADOE GSE Framework 8th Grade Unit 2 On your paper going down in a column, write 106, 105, 104, …, 100, 10-1, 10-2, 10-3. Simplify each. You may use a calculator if needed. If answer is given as a decimal, convert it to a fraction. Observations? Write table on the board.

Observations? If power is positive, the answers are whole numbers. If power is positive, the bigger the power, the bigger the answer. If power is negative, the answers are decimals or fractions. If power is positive, the power tells how many zeroes are in the answer. If you look at the answers going down the page, they are getting smaller. If you look at the answers going down the page, they are getting smaller by a factor of 10. Make sure to point out 10-1 = 0.1 = 1/10 = one-tenth 10-2 = 0.01 = 1/100 = one-hundredth = 1/102 10-3 = 0.001 = 1/1000 = one-thousandth = 1/103 etc

Observations? If you look at the answers going up the page, they are getting bigger. If you look at the answers going up the page, they are getting bigger by a factor of 10. Is 100 weird? It fits the pattern… If power is negative, the power tells how many decimal places are in the answer. If power is negative, the denominator is same as the answer for when the power is positive. Hmmm… Make sure to point out 10-1 = 0.1 = 1/10 = one-tenth 10-2 = 0.01 = 1/100 = one-hundredth = 1/102 10-3 = 0.001 = 1/1000 = one-thousandth = 1/103 etc

Exploring Powers of 2 Let’s see if those observations hold for powers of 2. On your paper going down in a column, write 26, 25, 24, …, 20, 2-1, 2-2, 2-3. Simplify each. You may use a calculator if needed. If answer is given as a decimal, convert it to a fraction. Observations? Write table on the board

Observations? If power is positive, the answers are whole numbers. If power is positive, the bigger the power, the bigger the answer. If power is negative, the answers are decimals or fractions. If you look at the answers going down the page, they are getting smaller. If you look at the answers going down the page, they are getting smaller by a factor of 2. Make sure to point out 2-1 = 0.5 = 1/2 = one-half 2-2 = 0.25 = 1/4 = one-fourth = 1/22 2-3 = 0.125 = 1/8 = one-eighth = 1/23 etc

Observations? If you look at the answers going up the page, they are getting bigger. If you look at the answers going up the page, they are getting bigger by a factor of 2. Is 20 weird? It fits the pattern… If power is negative, the power tells how many decimal places are in the answer. Does this hold for powers of 2 like it did for powers of 10? If power is negative, the denominator is same as the answer for when the power is positive. Hmmm… Make sure to point out 2-1 = 0.5 = 1/2 = one-half 2-2 = 0.25 = 1/4 = one-fourth = 1/22 2-3 = 0.125 = 1/8 = one-eighth = 1/23 etc

Observations? Can you come up with a rule that will always work? Negative Exponents Observations? Can you come up with a rule that will always work?

Negative Exponents Observations? Can you come up with a rule that will always work? In your own words? When you have a negative exponent, you can make it the reciprocal.

Zero Exponent Rule 100 = 1 20 = 1 50 = ? 990 = ? 100000 = ? 0.0050 = ? You may use a calculator.

Can you come up with a rule that will always work? Zero Exponent Rule 100 = 1 20 = 1 50 = 1 990 = 1 100000 = 1 0.0050 = 1 Observations? Can you come up with a rule that will always work? a0 = 1 In your own words? Anything to the zero power is 1.

Raising to the Zero and Negative Powers https://www.illustrativemathematics.org/content- standards/8/EE/A/1/tasks/1438 In this problem c represents a positive number. The quotient rule for exponents says that if m and n are positive integers with m > n, then

Raising to the Zero and Negative Powers A. What expression does the quotient rule provide for when m = n?

Raising to the Zero and Negative Powers

Raising to the Zero and Negative Powers B. If m = n, simplify without using the quotient rule.

Raising to the Zero and Negative Powers Anything divided by itself is 1.

Raising to the Zero and Negative Powers What do the previous two questions suggest is a good definition for c0? c0 = 1 (for any positive number c)

Raising to the Zero and Negative Powers D. What expression does the quotient rule provide for

Raising to the Zero and Negative Powers

Raising to the Zero and Negative Powers What expression do we get for if we use the value for c0 found in part C?

Raising to the Zero and Negative Powers

Raising to the Zero and Negative Powers Using parts D & E, propose a definition for the expression c – n.

Raising to the Zero and Negative Powers

Exponent Rules

Mathematics Assessment Project http://map.mathshell.org Tools for formative and summative assessment that make knowledge and reasoning visible, and help teachers to guide students in how to improve, and monitor their progress. Formative Assessment Lessons (FALs)  grades 6-11 This is a great site for tasks for middle grades and high school professional learning.

MAP FALs Lesson Plans – Concept development vs. Problem Solving Process – think independently & jot down ideas/work, work within a group to come up with a better/more efficient solution, discuss student solutions, & discuss provided student solutions PPT slides Worksheets

Applying Properties of Exponents http://map.mathshell.org/download.php?fileid=1668 Pre & post assessments Card sort

A: 8 × 4 = 32 B: 16 ÷ 8 = 2 C: 8 ÷ 16 = ½ D: 8 ÷ 8 = 1 Powers of 2 Write these as powers of 2. A: 8 × 4 = 32 B: 16 ÷ 8 = 2 C: 8 ÷ 16 = ½ Change all of these to powers of 2 & look at simplified solutions. D: 8 ÷ 8 = 1

Card Sort There are S cards and E cards. Select a card and find all other cards that have the same value as the one you have chosen. Most only have an S card and E card match. Some match to more than one E card or S card. *On a sheet of paper, you can write down the simplified solutions to help you match them.

Card Sort Solution S1 & E8 S2 & E9 S3 & E13 S4 & E4 S5 & E2 S6 & S10 & E10 & E12 S7 & E5 & E14 S8 & E1 & E7 S9 &E6 E3 & E11