Factors · Primes & prime factorization · 3.4

Exponent form.

When a prime repeats, write it with an exponent. 72 = 2×2×2×3×3 = 2³ × 3². Exponent form is the standard, compact way to write a prime factorization.

01 / THE IDEA

See it work.

Factor tree

Split until every leaf is prime.

The prime leaves are the prime factorization.

722362182933

72 = 2³ × 3²

Watch the method

Exponent form

72 = 2×2×2×3×3 = 2³ × 3².

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1

Build a factor tree

Split into factors until every leaf is prime.

2×2×2×3×32 \times 2 \times 2 \times 3 \times 3
2

Write the prime factorization

Collect the primes (use exponents for repeats).

72=23×3272 = 2^{3} \times 3^{2}
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Work it out.

Type primes, e.g. 2^3·3.

streak 0
best 0
… =
— / WHY IT WORKS

The idea underneath.

Compact with exponents

Count how many times each prime appears; that count is its exponent.

Why it works

Three 2s → 2³.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Write the prime factorization of 28.

1

Build a factor tree

Split into factors until every leaf is prime. 2 × 2 × 7

2

Write the prime factorization

Collect the primes (use exponents for repeats). 28 = 2^2 × 7

EX 2medium

Write the prime factorization of 28.

1

Build a factor tree

Split into factors until every leaf is prime. 2 × 2 × 7

2

Write the prime factorization

Collect the primes (use exponents for repeats). 28 = 2^2 × 7

EX 3hard

Write the prime factorization of 100.

1

Build a factor tree

Split into factors until every leaf is prime. 2 × 2 × 5 × 5

2

Write the prime factorization

Collect the primes (use exponents for repeats). 100 = 2^2 × 5^2

04 / COMMON TRAPS

Where students slip.

Trap

Writing the count wrong.

Fix

The exponent is how many times the prime appears.

Trap

Dropping a prime.

Fix

Every prime in the tree must appear.

05 / QUIZ

Test yourself.

The Test

12 problems across three tiers. Auto-graded on submit. Hints and a full walkthrough on every question. Your work is saved on this device — sign in to keep it across devices.

0/12Answered
How to answerType just the value. Formatting won't trip you up — we ignore spaces, capital letters, commas, and subscript style, so t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).
Core

Core (4)

C1

Write the prime factorization of 18.

Need a hint?
  • Count each prime’s copies.
  • Write that count as the exponent.
C2

Write the prime factorization of 60.

Need a hint?
  • Count each prime’s copies.
  • Write that count as the exponent.
C3

Write the prime factorization of 28.

Need a hint?
  • Count each prime’s copies.
  • Write that count as the exponent.
C4

Write the prime factorization of 20.

Need a hint?
  • Count each prime’s copies.
  • Write that count as the exponent.
Apply

Apply (4)

A1

Write the prime factorization of 18.

Need a hint?
  • Count each prime’s copies.
  • Write that count as the exponent.
A2

Write the prime factorization of 60.

Need a hint?
  • Count each prime’s copies.
  • Write that count as the exponent.
A3

Write the prime factorization of 16.

Need a hint?
  • Count each prime’s copies.
  • Write that count as the exponent.
A4

Write the prime factorization of 20.

Need a hint?
  • Count each prime’s copies.
  • Write that count as the exponent.
Extension

Extension (4)

E1

Write the prime factorization of 12.

Need a hint?
  • Count each prime’s copies.
  • Write that count as the exponent.
E2

Write the prime factorization of 54.

Need a hint?
  • Count each prime’s copies.
  • Write that count as the exponent.
E3

Write the prime factorization of 54.

Need a hint?
  • Count each prime’s copies.
  • Write that count as the exponent.
E4

Write the prime factorization of 24.

Need a hint?
  • Count each prime’s copies.
  • Write that count as the exponent.