Back to Lesson

Teacher Guide: Introduction to Exponents

Learn what exponents are and how they simplify repeated multiplication.

Present LessonShare to ClassroomView Student Page

Use this lesson with your class

Free, no student accounts needed.

Share with students

Students open the lesson and practise with instant feedback.

Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Exponents & Roots. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Define exponent and base in exponential notation
  • Evaluate expressions with positive integer exponents
  • Convert between repeated multiplication and exponential form
  • Apply exponents to real-world contexts including area and volume
Prerequisites
  • • Fluency with multiplication facts
  • • Understanding of repeated operations
  • • Familiarity with area of squares and volume of cubes
Discussion Starters
  • 1. Why do you think we invented exponents instead of just writing out all the multiplications?
  • 2. Where have you seen very large numbers in science or technology that might use exponents?
  • 3. If you fold a piece of paper in half 10 times, how many layers would you have? How does this relate to exponents?
  • 4. Why do you think computer memory uses powers of 2 instead of powers of 10?
Common Misconceptions

Exponentiation is repeated addition (thinking 34=3+3+3+33^4 = 3 + 3 + 3 + 3)

Remediation: Contrast clearly: 3×4=123 \times 4 = 12 (addition shortcut) vs 34=813^4 = 81 (multiplication shortcut). Use visual models showing groups.

The exponent multiplies the base (thinking 53=155^3 = 15)

Remediation: Write out the expanded form every time: 53=5×5×55^3 = 5 \times 5 \times 5. Count the 5s - there are three of them being multiplied.

Differentiation Ideas

For Struggling Students:

  • • Use manipulatives to show repeated multiplication visually
  • • Start with only squares (n2n^2) before introducing other exponents
  • • Create multiplication tables for powers of 2, 3, and 10

For On-Level Students:

  • • Compare powers with same base vs same exponent
  • • Convert large numbers to powers of 10
  • • Solve word problems involving area and volume

For Advanced Students:

  • • Explore patterns in units digits of powers
  • • Investigate negative bases (what is (−2)3(-2)^3?)
  • • Begin exploring exponential growth problems
Standards Alignment
  • 6.EE.A.1 (CCSS.MATH.CONTENT.6.EE.A.1)

    Write and evaluate numerical expressions involving whole-number exponents

  • 6.EE.A.2c (CCSS.MATH.CONTENT.6.EE.A.2c)

    Evaluate expressions at specific values of their variables

Lesson Resources
  • visualPower Tower Builder

    Build visual towers showing repeated multiplication

  • activityPowers of 2 Hunt

    Find powers of 2 in everyday technology

  • worksheetExponent Exploration

    Practice converting between forms and evaluating powers

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

An exponent tells you how many times a number is used as a factor in a multiplication.
In the expression 252^5:
  • 22 is the base (the number being multiplied)
  • 55 is the exponent (how many 2s are multiplied together)
25=2×2×2×2×2=322^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32
We read 252^5 as "two to the fifth power" or "two to the power of five."
Special names:
  • n2n^2 is called "n squared" (like the area of a square)
  • n3n^3 is called "n cubed" (like the volume of a cube)

Worked Examples

Calculate 343^4
1

Identify base and exponent

Base = 3, Exponent = 4 → Multiply four 3s together
2

Write out the multiplication

34=3×3×3×33^4 = 3 \times 3 \times 3 \times 3 → Four 3s multiplied
3

Multiply step by step

3×3=93 \times 3 = 9, then 9×3=279 \times 3 = 27, then 27×3=8127 \times 3 = 81

Common Mistakes

Multiplying base by exponent: 34=123^4 = 12

Why it's wrong: This confuses exponentiation with multiplication. The exponent tells you how many times to multiply, not the number to multiply by.

Correct: 34=3×3×3×3=813^4 = 3 \times 3 \times 3 \times 3 = 81, not 3×4=123 \times 4 = 12

Swapping base and exponent: treating 262^6 as 6×6=366 \times 6 = 36

Why it's wrong: The base (2) is the number you multiply. The small raised number (6) only counts how many 2s you write.

Correct: 26=2×2×2×2×2×2=642^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64, but 62=6×6=366^2 = 6 \times 6 = 36

Writing the wrong number of zeros: 105=1,000,00010^5 = 1{,}000{,}000

Why it's wrong: Each factor of 10 adds exactly one zero, so the exponent tells you how many zeros to write.

Correct: 105=10×10×10×10×10=100,00010^5 = 10 \times 10 \times 10 \times 10 \times 10 = 100{,}000: a 1 followed by 5 zeros

Why It Matters

Exponents are a powerful shorthand that appears everywhere:
  • Computer storage: Memory is measured in powers of 2 (1 KB = 2102^{10} bytes)
  • Science: Atoms are incredibly small (10−1010^{-10} meters)
  • Finance: Compound interest grows exponentially
  • Biology: Bacteria can double every hour (2242^{24}, more than 16 million in one day!)
Without exponents, writing 101210^{12} (one trillion) would require 12 zeros!

Real World Applications

Computer Memory

Computer memory is measured in powers of 2 because computers use binary (0s and 1s).

Example:

1 Kilobyte = 2102^{10} = 1,024 bytes. 1 Megabyte = 2202^{20} = 1,048,576 bytes.

1Try It Yourself

A small text file is 2122^{12} bytes (4 KB).

How many bytes is that exactly?

Step 1: Write the mathematical expression

Calculate 2122^{12}:

Bacterial Growth

Bacteria multiply by splitting in two. Each hour, the population doubles.

Example:

Starting with 1 bacterium, after 10 hours you have 2102^{10} = 1,024 bacteria!

2Try It Yourself

You start with 1 bacterium that doubles every hour.

How many bacteria after 8 hours?

Step 1: Write the mathematical expression

Calculate 282^8:

Area and Volume

Exponents naturally appear in geometry: area uses squared, volume uses cubed.

Example:

A square with side 7 cm has area 72=497^2 = 49 square cm. A cube with side 4 cm has volume 43=644^3 = 64 cubic cm.

3Try It Yourself

You have a cube-shaped box with sides of 5 cm.

What is the volume of the box?

Step 1: Write the mathematical expression

Calculate 535^3:

Key Takeaways

  • 1An exponent tells how many times the base is used as a factor: bn=b×b×...×bb^n = b \times b \times ... \times b (n factors)
  • 2The base is the number being multiplied; the exponent is how many times
  • 3n2n^2 is "n squared" and n3n^3 is "n cubed"
  • 4Powers of 10 are especially useful: 10n10^n is 1 followed by n zeros
  • 5Any number to the power of 1 equals itself: n1=nn^1 = n

Frequently Asked Questions

What is any number to the power of 0?

Any non-zero number to the power of 0 equals 1. For example, 50=15^0 = 1, 1000=1100^0 = 1. This will be covered in detail in a future lesson!

Why is it called 'squared' and 'cubed'?

'Squared' comes from calculating the area of a square (side times side). 'Cubed' comes from calculating the volume of a cube (side times side times side).

Is 232^3 the same as 323^2?

No! 23=82^3 = 8 (three 2s multiplied) but 32=93^2 = 9 (two 3s multiplied). The order matters!

Glossary

Exponent
The small number written above and to the right that shows how many times the base is used as a factor
Base
The number that is multiplied repeatedly in a power
Power
The result of raising a base to an exponent (e.g., 8 is a power of 2)
Squared
Raised to the power of 2 (multiplied by itself once)
Cubed
Raised to the power of 3 (multiplied by itself twice)

More in This Topic

Laws of Exponents

Intermediate25 min

Square Roots and Cube Roots

Intermediate25 min

Product Rule for Exponents

Intermediate20 min

Quotient Rule for Exponents

Intermediate20 min