Powers · Multiply — product rule · 3.1

Product rule.

To multiply powers of the SAME base, keep the base and ADD the exponents: xᵃ · xᵇ = xᵃ⁺ᵇ. It works because you are just counting all the factors. x²·x³ = (xx)(xxx) = x⁵.

01 / THE IDEA

See it work.

See the factors

Line up the factors and count.

Every power is just repeated multiplication.

xxxxx

Watch the method

Product rule

x²·x³ = x⁵ — count the factors.

1 / 2
x2⋅x3x^{2} \cdot x^{3}
1

Add the exponents

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

x2⋅x3=x5x^{2} \cdot x^{3} = x^{5}
2

Write the product

Same base when multiplying → add exponents.

x2⋅x3=x5x^{2} \cdot x^{3} = x^{5}
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Work it out.

Type with ^ for powers — e.g. x^5.

streak 0
best 0
… =
— / WHY IT WORKS

The idea underneath.

Count the factors

x² is two x’s, x³ is three x’s; multiplied together that’s five x’s = x⁵. Hence add the exponents.

Why it works

(xx)(xxx) = xxxxx.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Simplify x^4 · x^2

1

Add the exponents

Same base x — keep the base, add the exponents: 4 + 2. x^4 · x^2 = x^6

2

Write the product

Same base when multiplying → add exponents. x^4 · x^2 = x^6

EX 2medium

Simplify x^5 · x^3

1

Add the exponents

Same base x — keep the base, add the exponents: 5 + 3. x^5 · x^3 = x^8

2

Write the product

Same base when multiplying → add exponents. x^5 · x^3 = x^8

EX 3hard

Simplify y^2 · y^2

1

Add the exponents

Same base y — keep the base, add the exponents: 2 + 2. y^2 · y^2 = y^4

2

Write the product

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

04 / COMMON TRAPS

Where students slip.

Trap

Multiplying the exponents (getting x⁶).

Fix

Multiplying powers ADDS exponents; (xᵃ)ᵇ multiplies them.

Trap

Changing the base.

Fix

The base stays x.

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 x^2 · x^2

Need a hint?
  • Keep the base; add the exponents.
  • Count all the factors to check.
C2

Simplify x^3 · x^4

Need a hint?
  • Keep the base; add the exponents.
  • Count all the factors to check.
C3

Simplify x^4 · x^2

Need a hint?
  • Keep the base; add the exponents.
  • Count all the factors to check.
C4

Simplify x^4 · x^2

Need a hint?
  • Keep the base; add the exponents.
  • Count all the factors to check.
Apply

Apply (4)

A1

Simplify z^4 · z^3

Need a hint?
  • Keep the base; add the exponents.
  • Count all the factors to check.
A2

Simplify z^3 · z^2

Need a hint?
  • Keep the base; add the exponents.
  • Count all the factors to check.
A3

Simplify x^2 · x^4

Need a hint?
  • Keep the base; add the exponents.
  • Count all the factors to check.
A4

Simplify z^4 · z^4

Need a hint?
  • Keep the base; add the exponents.
  • Count all the factors to check.
Extension

Extension (4)

E1

Simplify y^4 · y^2

Need a hint?
  • Keep the base; add the exponents.
  • Count all the factors to check.
E2

Simplify x^2 · x^5

Need a hint?
  • Keep the base; add the exponents.
  • Count all the factors to check.
E3

Simplify z^4 · z^2

Need a hint?
  • Keep the base; add the exponents.
  • Count all the factors to check.
E4

Simplify z^2 · z^3

Need a hint?
  • Keep the base; add the exponents.
  • Count all the factors to check.