Powers · Combine the laws · 7.1

Pick the right law.

Each operation has its own rule: multiplying adds exponents, dividing subtracts, a power of a power multiplies, and a zero exponent gives 1. The skill is choosing the right one.

01 / THE IDEA

See it work.

Watch the method

Pick the right law

Read the operation, then apply its rule.

1 / 1
(x3)4(x^{3})^{4}
1

Multiply the exponents

A power of a power: keep the base, multiply 3 × 4.

(x3)4=x12(x^{3})^{4} = x^{12}
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
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— / WHY IT WORKS

The idea underneath.

The decision

Multiply → add exponents. Divide → subtract. Power of a power → multiply. Exponent 0 → 1. Negative exponent → reciprocal.

Why it works

Identify the operation first.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Simplify y^0

1

Zero exponent

Any non-zero base to the power 0 is 1 (it comes from x^a ÷ x^a). y^0 = 1

EX 2medium

Simplify z^5 ÷ z^1

1

Subtract the exponents

Same base z — keep the base, subtract: 5 − 1. z^5 ÷ z^1 = z^4

2

Write the quotient

Same base when dividing → subtract exponents. \fracz^5z^1 = z^4

EX 3hard

Simplify (y^4)^2

1

Multiply the exponents

A power of a power: keep the base, multiply 4 × 2. (y^4)^2 = y^8

04 / COMMON TRAPS

Where students slip.

Trap

Adding when you should multiply.

Fix

Multiplying powers adds; a power of a power multiplies.

Trap

Using the product rule on a quotient.

Fix

Dividing subtracts the exponents.

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 z^0

Need a hint?
  • Identify the operation first.
  • Then apply its exponent rule.
C2

Simplify y^0

Need a hint?
  • Identify the operation first.
  • Then apply its exponent rule.
C3

Simplify x^2 · x^4

Need a hint?
  • Identify the operation first.
  • Then apply its exponent rule.
C4

Simplify x^3 · x^3

Need a hint?
  • Identify the operation first.
  • Then apply its exponent rule.
Apply

Apply (4)

A1

Simplify y^5 ÷ y^2

Need a hint?
  • Identify the operation first.
  • Then apply its exponent rule.
A2

Simplify (x^3)^2

Need a hint?
  • Identify the operation first.
  • Then apply its exponent rule.
A3

Simplify (x^3)^4

Need a hint?
  • Identify the operation first.
  • Then apply its exponent rule.
A4

Simplify x^3 · x^5

Need a hint?
  • Identify the operation first.
  • Then apply its exponent rule.
Extension

Extension (4)

E1

Simplify z^0

Need a hint?
  • Identify the operation first.
  • Then apply its exponent rule.
E2

Simplify y^4 ÷ y^2

Need a hint?
  • Identify the operation first.
  • Then apply its exponent rule.
E3

Simplify y^0

Need a hint?
  • Identify the operation first.
  • Then apply its exponent rule.
E4

Simplify x^3 ÷ x^2

Need a hint?
  • Identify the operation first.
  • Then apply its exponent rule.