Powers · Multiply — product rule · 3.3

Numbers, same base.

The product rule works for numeric bases too: 2³·2² = 2³⁺² = 2⁵ = 32. Keep the base, add the exponents, then evaluate.

01 / THE IDEA

See it work.

Watch the method

Numbers, same base

2³·2² = 2⁵ = 32.

1 / 2
23⋅222^{3} \cdot 2^{2}
1

Same base — add the exponents

Keep the base 2, add 3 + 2.

23⋅22=252^{3} \cdot 2^{2} = 2^{5}
2

Evaluate

Now compute the power.

25=322^{5} = 32
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Work it out.

Type with ^ for powers — e.g. x^5.

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— / WHY IT WORKS

The idea underneath.

Same idea, numbers

Equal bases multiply by adding exponents; afterwards you can evaluate the single power.

Why it works

2³·2² is five 2s = 32.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Evaluate 3^3 · 3^1

1

Same base — add the exponents

Keep the base 3, add 3 + 1. 3^3 · 3^1 = 3^4

2

Evaluate

Now compute the power. 3^4 = 81

EX 2medium

Evaluate 2^2 · 2^1

1

Same base — add the exponents

Keep the base 2, add 2 + 1. 2^2 · 2^1 = 2^3

2

Evaluate

Now compute the power. 2^3 = 8

EX 3hard

Evaluate 5^3 · 5^1

1

Same base — add the exponents

Keep the base 5, add 3 + 1. 5^3 · 5^1 = 5^4

2

Evaluate

Now compute the power. 5^4 = 625

04 / COMMON TRAPS

Where students slip.

Trap

Multiplying the bases (getting 4⁵).

Fix

Keep the base; only add exponents.

Trap

Multiplying the exponents.

Fix

They add under multiplication.

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

Evaluate 3^2 · 3^2

Need a hint?
  • Keep the base; add the exponents.
  • Then evaluate the power.
C2

Evaluate 5^3 · 5^1

Need a hint?
  • Keep the base; add the exponents.
  • Then evaluate the power.
C3

Evaluate 3^3 · 3^2

Need a hint?
  • Keep the base; add the exponents.
  • Then evaluate the power.
C4

Evaluate 5^3 · 5^1

Need a hint?
  • Keep the base; add the exponents.
  • Then evaluate the power.
Apply

Apply (4)

A1

Evaluate 5^2 · 5^2

Need a hint?
  • Keep the base; add the exponents.
  • Then evaluate the power.
A2

Evaluate 3^2 · 3^2

Need a hint?
  • Keep the base; add the exponents.
  • Then evaluate the power.
A3

Evaluate 3^3 · 3^2

Need a hint?
  • Keep the base; add the exponents.
  • Then evaluate the power.
A4

Evaluate 2^2 · 2^1

Need a hint?
  • Keep the base; add the exponents.
  • Then evaluate the power.
Extension

Extension (4)

E1

Evaluate 5^3 · 5^2

Need a hint?
  • Keep the base; add the exponents.
  • Then evaluate the power.
E2

Evaluate 2^3 · 2^1

Need a hint?
  • Keep the base; add the exponents.
  • Then evaluate the power.
E3

Evaluate 2^2 · 2^1

Need a hint?
  • Keep the base; add the exponents.
  • Then evaluate the power.
E4

Evaluate 2^2 · 2^1

Need a hint?
  • Keep the base; add the exponents.
  • Then evaluate the power.