Exponents

Videos and solutions to help Grade 6 students learn what are exponents and how to use exponents.

New York State Common Core Math Grade 6, Module 4, Lesson 5

Worksheets for Grade 6

Lesson 5 Student Outcomes

Students discover that 3x = x + x + x is not the same thing as x³ which is x • x • x.

Students understand that a base number can be represented with a positive whole number, positive fraction, or positive decimal and that for any number b, we define b^m to be m factors of b where b is the base and m is called the exponent or power of b.

Opening Exercise

As you evaluate these expressions, pay attention to how you arrive at your answers.

4 + 4 + 4 + 4 + 4 + 4 + 4 +4 + 4 + 4
9 + 9 + 9 + 9 + 9
10 + 10 + 10 + 10 + 10

Multiplication is a faster way to add numbers when the addends are the same.

When we add five groups of 10, we use an abbreviation and a different notation, called multiplication.
10 + 10 + 10 + 10 + 10 = 10 × 5 = 50.

If multiplication is a more efficient way to represent addition problems involving the repeated addition of the same addend, do you think there might be a more efficient way to represent the repeated multiplication of the same factor, as in 10 x 10 x 10 x 10 x 10.

When we add 5 groups of 10, we write 5 × 10, but when we multiply 5 copies of 10, we write 10⁵. So, multiplication by 5 in the context of addition corresponds exactly to the exponent in the context of multiplication.

The repeated factor is called the base and the exponent is also called the power.

There is a special name for numbers raised to the second power. When a number is raised to the second power, it is called squared.

There is also a special name for numbers raised to the third power. When a number is raised to the third power, it is called cubed.

Examples
Write each expression in exponential form.

1. 5 × 5 × 5 × 5 × 5
2. 2 × 2 × 2 × 2
   Write each expression in expanded form.
3. 8³
4. 10⁶
5. g³

What is the difference between 3g and g³?

The base number can be written in decimal or fraction form.
(3.8)⁴
(2/3)²

Exercises

1. Fill in the missing expressions for each row. For whole number and decimal bases, use a calculator to find the standard form of the number. For fraction bases, leave your answer as a fraction.
2. Write “five cubed” in all three forms: exponential form, written as a series of products, standard form.
3. Write “fourteen and seven tenths squared” in all three forms.
4. One student thought two to the third power was equal to six. What mistake do you think they made and how would you help them fix their mistake?

Lesson Summary

Exponential Notation for Whole Number Exponents: Let be a non-zero whole number. For any number b, the expression b^m is the product of m factors of b.

The number b is called the base, and m is called the exponent or power of b. When m is 1, “the product of one factor of ” just means b, i.e. b¹ = b. Raising any non-zero number to the power of 0 is defined to be 1, i.e., b⁰ = 1 for all b ≠ 0.

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Lesson 5 Examples and Exercises

Examples 1 - 5
Go back to Examples 1ִ and use a calculator to evaluate the expressions.

Example 6
3.8⁴

Example 7
2.1 × 2.1

Example 8
0.75 × 0.75 × 0.75

The base number can also be a fraction. Convert the decimals to fractions in Examples 7 and 8 and evaluate. Leave your answer as a fraction.

Example 9
1/2 × 1/2 × 1/2

Example 10
(2/3)²

Exercises 1 - 4

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Lesson 5 Exit Ticket

1. What is the difference between 6z and z⁶?
2. Write 10 ³as a multiplication expression having repeated factors.
3. Write 8 × 8 × 8 × 8 using exponents.

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Problem Set

1. Complete the table by filling in the blank cells. Use a calculator when needed. Exponential Form, Expanded Form, Standard Form
2. Why do whole numbers raised to an exponent get greater, while fractions raised to an exponent get smaller? 3. The powers of 2 that are in the range 2 through 1,000 are 2, 4, 8, 16, 32, 64, 128, 256, and 512. Find all the powers of 3 that are in the range 3 through 1,000.
3. Find all the powers of 4 in the range 4 through 1,000.
4. Write an equivalent expression for n × a using only addition.
5. Write an equivalent expression for w^b using only multiplication.
   a. Explain what w is in this new expression.
   b. Explain what b is in this new expression.
6. What is the advantage of using exponential notation?
7. What is the difference between 4x and x⁴? Evaluate both of these expressions when x = 2.

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