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Glacius
FoundationsConcept reference

The zero exponent

The zero power of a nonzero real number is one.

1 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. Moving down one exponent divides by the base
  3. Zero must be the exponent
  4. Check your understanding
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections

01Moving down one exponent divides by the base#

Moving down one exponent divides by the base. For base 3, the values go from 9 to 3 to 1.

The same rule y=3^x gives 9 at exponent 2, 3 at exponent 1 and 1 at exponent 0. Each leftward unit step divides by 3.The same rule y=3^x gives 9 at exponent 2, 3 at exponent 1 and 1 at exponent 0. Each leftward unit step divides by 3.
Figure 1The same rule y=3^x gives 9 at exponent 2, 3 at exponent 1 and 1 at exponent 0. Each leftward unit step divides by 3.
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The value at exponent zero keeps that pattern intact. For every nonzero base aa:

a0=1a^0=1
Check your reasoning

Evaluate (−9)0(-9)^0.

Show answer and explanation
1

The whole base −9 is nonzero. Its zero power is 1.

02Zero must be the exponent#

Zero must be the exponent. In 030^3, zero is the base: three factors of zero give zero. In 303^0, the nonzero base has exponent zero, giving 1.

The condition matters. This rule does not assign a value to 000^0. Keep the nonzero-base condition whenever you use it.

Check your reasoning

A learner writes 00=10^0=1. What does this lesson’s rule justify?

  1. AIt gives 1.
  2. BIt gives 0.
  3. CIt does not apply.
Show answer and explanation
It does not apply.

The rule only covers a nonzero base.

03Check your understanding#

Check your reasoning

A growth table multiplies its starting amount by 505^0 at the start. What is that multiplier?

Show answer and explanation
1

The starting multiplier is 1, so the starting amount is unchanged.

Key takeaway

A zero exponent gives 1 when the base is nonzero.

  • Evaluate a nonzero base raised to the zero power.

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “The zero exponent.” Math behind ML. /learn/f-power-zero