Powers · Multiply — product rule · 3.2

With coefficients.

When the terms have coefficients, multiply the coefficients and add the exponents separately. 3x² · 4x³ = (3·4)x²⁺³ = 12x⁵.

01 / THE IDEA

See it work.

Watch the method

With coefficients

3x² · 4x³ = 12x⁵.

1 / 3
3x2⋅4x33x^{2} \cdot 4x^{3}
1

Multiply the coefficients

3 × 4 = 12.

3×4=123 \times 4 = 12
2

Add the exponents

Same base x — keep the base, add the exponents: 2 + 3.

x2⋅x3=x5x^{2} \cdot x^{3} = x^{5}
3

Write the product

Same base when multiplying → add exponents.

3x2⋅4x3=12x53x^{2} \cdot 4x^{3} = 12 x^{5}
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.

Two jobs

Coefficients multiply (numbers), exponents add (product rule). Do each separately.

Why it works

3·4 = 12 out front; x²·x³ = x⁵.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Simplify 4y^2 · 3y^1

1

Multiply the coefficients

4 × 3 = 12. 4 × 3 = 12

2

Add the exponents

Same base y — keep the base, add the exponents: 2 + 1. y^2 · y^1 = y^3

3

Write the product

Same base when multiplying → add exponents. 4y^2 · 3y^1 = 12 y^3

EX 2medium

Simplify 3z^3 · 3z^3

1

Multiply the coefficients

3 × 3 = 9. 3 × 3 = 9

2

Add the exponents

Same base z — keep the base, add the exponents: 3 + 3. z^3 · z^3 = z^6

3

Write the product

Same base when multiplying → add exponents. 3z^3 · 3z^3 = 9 z^6

EX 3hard

Simplify 4x^3 · 5x^1

1

Multiply the coefficients

4 × 5 = 20. 4 × 5 = 20

2

Add the exponents

Same base x — keep the base, add the exponents: 3 + 1. x^3 · x^1 = x^4

3

Write the product

Same base when multiplying → add exponents. 4x^3 · 5x^1 = 20 x^4

04 / COMMON TRAPS

Where students slip.

Trap

Adding the coefficients.

Fix

Coefficients multiply when you multiply terms.

Trap

Multiplying the exponents.

Fix

Exponents 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

Simplify 5x^1 · 3x^1

Need a hint?
  • Multiply the numbers in front.
  • Add the exponents.
C2

Simplify 2x^1 · 4x^1

Need a hint?
  • Multiply the numbers in front.
  • Add the exponents.
C3

Simplify 2y^1 · 3y^4

Need a hint?
  • Multiply the numbers in front.
  • Add the exponents.
C4

Simplify 3z^1 · 5z^2

Need a hint?
  • Multiply the numbers in front.
  • Add the exponents.
Apply

Apply (4)

A1

Simplify 5x^3 · 5x^4

Need a hint?
  • Multiply the numbers in front.
  • Add the exponents.
A2

Simplify 2z^2 · 2z^1

Need a hint?
  • Multiply the numbers in front.
  • Add the exponents.
A3

Simplify 5z^1 · 4z^1

Need a hint?
  • Multiply the numbers in front.
  • Add the exponents.
A4

Simplify 3y^3 · 5y^1

Need a hint?
  • Multiply the numbers in front.
  • Add the exponents.
Extension

Extension (4)

E1

Simplify 5y^4 · 5y^1

Need a hint?
  • Multiply the numbers in front.
  • Add the exponents.
E2

Simplify 3z^1 · 2z^1

Need a hint?
  • Multiply the numbers in front.
  • Add the exponents.
E3

Simplify 2z^3 · 4z^2

Need a hint?
  • Multiply the numbers in front.
  • Add the exponents.
E4

Simplify 5x^1 · 5x^1

Need a hint?
  • Multiply the numbers in front.
  • Add the exponents.