Exponent Rules Standard 8 EE 1 Today What

Exponent Rules Standard: 8. EE. 1

Today • What are Exponents? • How do you write an expression using exponents? • How do you evaluate an expression that contains exponents? • What RULES do we have for dealing with exponents?

What is an Exponent? Exponents are a short-cut way of writing repeated multiplication.

How do you Write an expression using exponents? Expressions can be written in two different forms. Factored Form Exponential Form 8 • 2 • 2 8² • 2³ 5 • 9 • 5 • 5 5⁴ • 9

How do you Evaluate an expression that contains exponents? To find 54 on my calculator I type in: = 625

How do you Evaluate an expression that contains exponents? 4 To find 3 • 2⁵ on my calculator I type in: = 2, 592

Write the expression in Exponential Form. 6 • 3 • 4 • 6 • 3 = 6³ • 3² • 4 Now, evaluate your expression using the ^ button on your calculator. 7, 776

When evaluating exponents you must watch the sign and the parenthesis! 5² = 5 • 5 = 25 – 5² = – (5 • 5) = – 25 (– 5)² = (– 5) • (– 5) = 25

Use your calculator. Be sure to enter the ( ) where indicated. Evaluate. 1. (– 4)² 16 2. – 4² -16 3. –(4)² -16 4. 4² 16

Multiplying Powers with the Same Base RULE 1

8 -37. Complete the table below. Expand each expression into factored form and then rewrite it in a simplified exponential form as shown in the example.

8 -37 a. Work with your team to compare the bases and exponents of the original form to the base and exponent of the simplified exponent form. Write a statement to describe the relationships you see. Original Form Simplified Form 52 • 5⁵ = 5⁷ The base is the same in both forms. 2² • 2⁴ = 26 The simplified exponent can be determined by adding (+) the original exponents together. 3⁷ • 3² = 3⁹

RULE for Multiplying Powers with the Same Base Step 1 Keep the Base Step 2 ADD the Exponents Step 3 Evaluate if needed 52 • 5⁵ = 5²⁺⁵ = 5⁷ or 78, 125 2² • 2⁴ = 2²⁺⁴ = 26 or 64 3⁷ • 3² = 3⁷⁺² = 3⁹ or 19, 683

Step 1 Keep the Base Step 2 ADD the Exponents Step 3 Evaluate if needed 8 -37 b. Write the expression in simplified exponential form. Use your new exponent rule! You do not have to evaluate it. (20¹² )(20⁵¹) = 20¹²⁺⁵¹ = 20⁶³

8 -37 c. One study team rewrote the expression 10³ • 5⁴ as 50⁷. Is their simplification correct? Explain your reasoning. NO it is not correct. 103 · 54 = 625, 000 , while 507 = 781, 250, 000 The BASES have to be the SAME! On this one… One of the bases is 10 and the other base is 5.

Find SET 1 on your practice sheet. You will have a set amount of time to complete these problems. Then we will review them.

Dividing Powers with the Same Base RULE 2

If you ADD (+) the exponents when multiplying powers with the same base…. What do you think you are supposed to do when dividing powers with the same base?

RULE for Dividing Powers with the Same Base Step 1 Keep the Base • Step 2 Subtract the Exponents Step 3 Evaluate if needed

Why does subtracting the exponents work? 5 6 2 6 = 6 • 6 • 6 = 3 6

Step 1 Keep the Base Step 2 Subtract the Exponents Step 3 Evaluate if necessary Write the expression in simplified exponential form. Use your new exponent rule! You do not have to evaluate them. = 7³⁻² = 7 = 9⁵⁻¹ = 9⁴ = 2⁹⁻⁴ = 2⁵

Find SET 2 on your practice sheet. You will have a set amount of time to complete these problems. Then we will review them.

Power of a Power RULE 3

8 -61. When a number is raised to a power, and then raised to a power again, the result follows a consistent pattern. Complete the table below. Expand each expression into factored form and then rewrite it with new exponents as shown in the example.

8 -61 a. Work with your team to describe the pattern between the exponents in the original form and the exponents in the simplified exponential form. Original Form Simplified Form 2 (5 )⁵ = 5¹⁰ (2²)⁴ = 2⁸ (3⁷)² = 3¹⁴ The exponents in the simplified form are the product of the exponent inside the parentheses and the exponent outside the parentheses.

RULE for a Power of a Power Step 1 Multiply the exponent inside the ( ) by the exponent on the outside. × (52)⁵ = 5¹⁰ = 9, 765, 625 × (2²)⁴ = 2⁸ = 256 × (3⁷)² = 3¹⁴ = 4, 782, 969 Step 2 Evaluate if needed

Visualize (20³)⁸ written in factored form. What would it look like? How many 20’s would be written down? 8 -61 b. the factored form would look like 8 sets of 20 · 20. There would be 24 of the 20’s. Now, write the express (20³)⁸ in simplified exponential form. You do not have to evaluate it. Use your new rule! × (20³)⁸ = 20²⁴

Find SET 3 on your practice sheet. You will have a set amount of time to complete these problems. Then we will review them.

Look at this example of how to find a Power of a Product. × (2² • 3³)⁵ = 2¹⁰ • 3¹⁵ Write this in simplified Exponential form. (7⁴ • 2³)² = 7⁸ • 2⁶

Find SET 4 on your practice sheet. You will have a set amount of time to complete these problems. Then we will review them.