Powers · Zero & negative exponents · 6.1

Zero exponent.

Any non-zero base to the power 0 equals 1. It follows from the quotient rule: xᵃ ÷ xᵃ = xᵃ⁻ᵃ = x⁰, but it is also xᵃ/xᵃ = 1. So x⁰ = 1.

01 / THE IDEA

See it work.

Watch the method

Zero exponent

x⁰ = 1 (from xᵃ ÷ xᵃ).

1 / 1
x0x^{0}
1

Zero exponent

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

x0=1x^{0} = 1
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.

Why it’s 1

xᵃ divided by itself is 1, and the quotient rule makes it x⁰ — so x⁰ must equal 1.

Why it works

x⁵/x⁵ = 1 = x⁰.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Simplify z^0

1

Zero exponent

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

EX 2medium

Simplify x^0

1

Zero exponent

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

EX 3hard

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

04 / COMMON TRAPS

Where students slip.

Trap

Saying x⁰ = 0.

Fix

x⁰ = 1, not 0.

Trap

Thinking 0⁰ is fine.

Fix

The rule is for non-zero bases.

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?
  • Any non-zero base to the 0 is 1.
  • It comes from xᵃ ÷ xᵃ.
C2

Simplify x^0

Need a hint?
  • Any non-zero base to the 0 is 1.
  • It comes from xᵃ ÷ xᵃ.
C3

Simplify y^0

Need a hint?
  • Any non-zero base to the 0 is 1.
  • It comes from xᵃ ÷ xᵃ.
C4

Simplify y^0

Need a hint?
  • Any non-zero base to the 0 is 1.
  • It comes from xᵃ ÷ xᵃ.
Apply

Apply (4)

A1

Simplify y^0

Need a hint?
  • Any non-zero base to the 0 is 1.
  • It comes from xᵃ ÷ xᵃ.
A2

Simplify y^0

Need a hint?
  • Any non-zero base to the 0 is 1.
  • It comes from xᵃ ÷ xᵃ.
A3

Simplify x^0

Need a hint?
  • Any non-zero base to the 0 is 1.
  • It comes from xᵃ ÷ xᵃ.
A4

Simplify z^0

Need a hint?
  • Any non-zero base to the 0 is 1.
  • It comes from xᵃ ÷ xᵃ.
Extension

Extension (4)

E1

Simplify y^0

Need a hint?
  • Any non-zero base to the 0 is 1.
  • It comes from xᵃ ÷ xᵃ.
E2

Simplify x^0

Need a hint?
  • Any non-zero base to the 0 is 1.
  • It comes from xᵃ ÷ xᵃ.
E3

Simplify z^0

Need a hint?
  • Any non-zero base to the 0 is 1.
  • It comes from xᵃ ÷ xᵃ.
E4

Simplify z^0

Need a hint?
  • Any non-zero base to the 0 is 1.
  • It comes from xᵃ ÷ xᵃ.