Write and Evaluate Expressions with Exponents

Lesson 1 of 1: Expressions and Equations

In this lesson:

• Write repeated multiplication using exponential notation
• Evaluate expressions with whole-number exponents
• Distinguish base from exponent to avoid computation errors

Learning Objectives for This Lesson

By the end of this lesson, you should be able to:

1. Write repeated multiplication using exponential notation — identify the base and exponent
2. Evaluate expressions with whole-number exponents, including perfect squares and perfect cubes
3. Distinguish between the base and the exponent; avoid multiplying base × exponent

A Better Way to Write Repeated Products

• You already know how to compute 2 × 2 × 2 × 2 × 2 = 32
• What if you needed to write 2 multiplied by itself twenty times?
• Mathematicians created a compact shorthand for this: exponential notation

Base and Exponent: Reading the Notation

Base = the repeated factor | Exponent = how many times it appears

Writing Repeated Multiplication in Exponent Form

Count the factors — that number becomes the exponent:

• 5 × 5 × 5 = (three 5s; "5 cubed")
• 2 × 2 × 2 × 2 = (four 2s; "2 to the fourth")
• 10 × 10 = (two 10s; "10 squared")

Expanding Exponent Notation into Repeated Factors

• = 4 × 4 × 4 = 64
• = 10 × 10 × 10 × 10 = 10,000
• = 1 × 1 × 1 × 1 × 1 × 1 = 1

Exponent Counts Factors, Not a Multiplier

• The exponent tells you how many factors of the base to write
• = 2 × 2 × 2 = 8 ← three factors of 2
• ← wrong: exponent is not a multiplier
• Rule: always expand first, then multiply

Quick Check: Writing and Expanding Exponents

Answer both before advancing:

1. Write 7 × 7 × 7 × 7 using exponent notation. What is the base? What is the exponent?
2. Expand as repeated multiplication. How many factors of 3 are there?

Pause — work it out, then advance to check.

Check-In Answers: Writing and Expanding

1. 7 × 7 × 7 × 7 = — base is 7, exponent is 4

2. = 3 × 3 × 3 × 3 — exponent 4 counts four factors of 3

From Writing to Evaluating: Computing the Value

• So far: writing repeated products as and expanding into repeated factors
• Next: evaluating — computing the numerical value of an exponential expression
• Method: expand into repeated factors first, then multiply left to right

Evaluating Exponential Expressions Step by Step

Procedure:

1. Identify the base and the exponent
2. Expand: write the base as a repeated multiplication
3. Multiply left to right, recording each partial product

Quick example: → 4 × 4 × 4 → 16 × 4 = 64

Evaluating Expressions with Partial Products

: 3 × 3 = 9 → × 3 = 27 → × 3 = 81

: 5 × 5 = 25

: 2 × 2 = 4 → × 2 = 8 → × 2 = 16 → × 2 = 32 → × 2 = 64

Powers Reference Table: Squares, Cubes, and Tens

Pattern for powers of 10: the exponent equals the number of zeros.

Special Cases: Exponent One and Exponent Zero

• Any base to the first power equals itself: (one factor)
• Any base to the zero power equals 1: (explained in Grade 8)
• Geometric link: = area of a square with side ; = volume of a cube

Quick Check: Evaluate and Spot the Error

Evaluate both expressions, showing your expanded form:

1. = ?
2. = ?

Spot the error: A student claims . What is the correct value, and what mistake did the student make?

Work it out before advancing.

Check-In Answers: Evaluate and Spot the Error

1. : 4 × 4 = 16, then 16 × 4 = 64
2. : 10 × 10 = 100

Error correction: , not 30. The mistake: multiplied 10 × 3 instead of expanding three factors of 10.

Key Takeaways and Common Mistakes to Avoid

✓ Exponent = how many times the base appears as a factor
✓ Evaluate: expand into repeated multiplication, multiply step by step
✓ Squared = exponent 2; cubed = exponent 3

Watch out: , not 6 — expand first; don't multiply base × exponent
Watch out: , not 12 — multiply factors, don't add them
Watch out: , not 30 — expand three factors of 10

Coming Up Next: Variables in Expressions

• Next lesson: 6.EE.A.2 — evaluating algebraic expressions with variables
• You'll apply these same skills when computing for a cube's volume
• The notation and evaluation procedure carry forward directly