Write and Evaluate Expressions with Exponents

In this lesson:

• Read and write exponential notation
• Evaluate expressions step by step
• Connect squared and cubed to area and volume

What You Will Be Able to Do

By the end, you will:

1. Name the base and exponent in any expression
2. Write repeated multiplication as an exponent
3. Expand and evaluate any exponential expression
4. Use "squared" and "cubed" for area and volume
5. Apply the rules: and

Eight Twos — Count the Factors

• Count the 2s. How many are there?
• Compute the product. What is it?
• What if there were twenty 2s?

Tedious to write and count — there must be a shorter way.

Reading the Notation: Base and Exponent

Reading Notation in Both Directions

Expanded → exponential:

Exponential → expanded:

The exponent is the count of factors — not a multiplier of the base.

Which Expression Does Represent?

Two expressions — one is what the notation means, one is not.

Which one does stand for? Commit before advancing.

Means Three Factors of Two

The exponent counts how many times the base appears — not a multiplier.

Squared and Cubed: Geometric Meaning

• "Squared" → exponent 2 → area of a square
• "Cubed" → exponent 3 → volume of a cube

Special Cases: Exponent 1 and Exponent 0

Exponent = 1: one factor of the base — no multiplication needed

Exponent = 0: any nonzero base to the zero power equals 1

Note: is not defined; all our examples use nonzero bases.

Check-In: Practice Translation Both Ways

Try both directions before the next slide.

1. Write in exponential form.
2. Expand as a repeated multiplication.

Pause and write your answers.

Why Exponentiation Is Its Own Operation

Three Steps to Evaluate Any Expression

Step 1 — Identify: name the base and the exponent

Step 2 — Expand: write out all the factors

Step 3 — Multiply: compute left to right, tracking partial products

Write the expanded form every time — it's the safeguard against wrong answers.

Worked Example: Using the Protocol on

Step 1: Base = 3, Exponent = 4

Step 2: Expand

Step 3: Multiply left to right

Worked Examples: and

(five squared):

(two to the sixth):

Six multiplications — tracking partial products prevents errors.

Your Turn: Evaluating Together

Step 1: Base = 10, Exponent = 3

Step 2: Expand — you write the three factors

Step 3: Multiply — compute the product

How many zeros does your answer have after the 1?

Watch Out: Is Not

Which equals 1,000?

• A.
• B.

Check: does 30 have three zeros after a 1? Does 1,000?

Worked Examples: and

Fractional base — :

Base of 1 — :

The protocol works for any base — whole numbers, fractions, or 1.

Check-In: Evaluate — Show All Three Steps

Write your work:

1. Base = ___, Exponent = ___
2. Expand: $4^3 = $ ___ × ___ × ___
3. Multiply: = ___

Do this one yourself before advancing.

Fill In the Benchmark Powers Table

Expand first, then multiply. Answers on the next slide.

Benchmark Powers: Completed Reference Table

Formula Bridge: Volume of a Cube

A cube with side cm:

The formula uses exponents — evaluation is the skill you just practiced.

Preview: Standard 6.EE.A.2 will give you different values of to substitute.

A Surprising Result — Coincidence or Pattern?

Evaluate both expressions using the three-step protocol:

Do both before advancing. Are the results the same? Why?

Practice Evaluating Three Expressions Independently

Show all three steps for each:

Expand first, then multiply. Write partial products.

Answers and Common Errors to Avoid

Common error: (multiplied 2×5 instead of expanding five 2s)

Find the Error in This Evaluation

A student writes:

What went wrong? Identify the error before advancing.

Why , Not 12

The student added. The exponent calls for multiplication.

• Repeated addition →
• Repeated multiplication →

The magnitude difference — 12 vs. 81 — is the check.

The Compression Hierarchy You Now Own

• Base = repeated factor; exponent = count of factors
• Protocol: identify → expand → multiply

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Next Lesson: Evaluating Formulas with Exponents

You can now evaluate any expression like or .

In Standard 6.EE.A.2, you'll evaluate expressions inside formulas:

You'll substitute a value for and apply today's protocol.