Powers · Multiply — product rule · 3.4

Several variables.

With more than one variable, apply the product rule to each base separately. x²y · x³y⁴ = x²⁺³ y¹⁺⁴ = x⁵y⁵.

01 / THE IDEA

See it work.

Watch the method

Several variables

x²y · x³y⁴ = x⁵y⁵.

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x2y⋅x3y4x^{2}y \cdot x^{3}y^{4}
1

Group each base

Collect the powers of each base separately.

x2y⋅x3y4x^{2}y \cdot x^{3}y^{4}
2

Add the exponents per base

For each base, add the exponents.

=x5y5= x^{5} y^{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.

Base by base

Collect the x’s and the y’s separately, adding each base’s exponents.

Why it works

x’s with x’s, y’s with y’s.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Simplify x^2y^3 · x^3y^2

1

Group each base

Collect the powers of each base separately. x^2y^3 · x^3y^2

2

Add the exponents per base

For each base, add the exponents. = x^5 y^5

EX 2medium

Simplify x^3y^2 · x^3y^2

1

Group each base

Collect the powers of each base separately. x^3y^2 · x^3y^2

2

Add the exponents per base

For each base, add the exponents. = x^6 y^4

EX 3hard

Simplify x^3y^2 · x^3y^2

1

Group each base

Collect the powers of each base separately. x^3y^2 · x^3y^2

2

Add the exponents per base

For each base, add the exponents. = x^6 y^4

04 / COMMON TRAPS

Where students slip.

Trap

Mixing the x and y exponents.

Fix

Add exponents within each base only.

Trap

Forgetting y = y¹.

Fix

A bare variable has exponent 1.

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^3y^1 · x^2y^3

Need a hint?
  • Handle each base separately.
  • Add the exponents per base.
C2

Simplify x^2y^3 · x^2y^1

Need a hint?
  • Handle each base separately.
  • Add the exponents per base.
C3

Simplify x^2y^2 · x^3y^1

Need a hint?
  • Handle each base separately.
  • Add the exponents per base.
C4

Simplify x^3y^2 · x^1y^1

Need a hint?
  • Handle each base separately.
  • Add the exponents per base.
Apply

Apply (4)

A1

Simplify x^2y^1 · x^2y^2

Need a hint?
  • Handle each base separately.
  • Add the exponents per base.
A2

Simplify x^2y^2 · x^1y^2

Need a hint?
  • Handle each base separately.
  • Add the exponents per base.
A3

Simplify x^1y^2 · x^3y^1

Need a hint?
  • Handle each base separately.
  • Add the exponents per base.
A4

Simplify x^2y^3 · x^2y^3

Need a hint?
  • Handle each base separately.
  • Add the exponents per base.
Extension

Extension (4)

E1

Simplify x^3y^3 · x^3y^2

Need a hint?
  • Handle each base separately.
  • Add the exponents per base.
E2

Simplify x^3y^2 · x^1y^1

Need a hint?
  • Handle each base separately.
  • Add the exponents per base.
E3

Simplify x^1y^2 · x^2y^3

Need a hint?
  • Handle each base separately.
  • Add the exponents per base.
E4

Simplify x^2y^3 · x^2y^2

Need a hint?
  • Handle each base separately.
  • Add the exponents per base.