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Powers

A positive whole-number power is the product of the stated number of copies of its base.

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On this page 8 sections
  1. Overview
  2. A positive whole-number exponent counts repeated factors
  3. Check your understanding
  4. Check your understanding
  5. Key takeaway
  6. Further questions
  7. Sources & further reading
  8. Concept connections

01A positive whole-number exponent counts repeated factors#

A positive whole-number exponent counts repeated factors. In 232^3, the base is 22 and the exponent is 33: multiply three copies of 22.

The notation 2 cubed expands to three factors of 2, whose product is 8.The notation 2 cubed expands to three factors of 2, whose product is 8.
Figure 1The notation 2 cubed expands to three factors of 2, whose product is 8.
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Work through the repeated product: 2×2=42\times2=4, then 4×2=84\times2=8. The exponent does not multiply the base. Three copies of 22 multiplied together differ from 2×32\times3.

23=2×2×2=82^3=2\times2\times2=8

Parentheses show exactly what is repeated. In (−2)3(-2)^3, every factor is −2-2, so the product is −8-8. With two negative factors, (−2)2=4(-2)^2=4. An exponent of 11 gives one copy of the base.

Check your reasoning

Calculate 333^3.

Show answer and explanation
27

Three factors of 3 give 3×3×3=273\times3\times3=27.

02Check your understanding#

Check your reasoning

Which expansion equals (−3)2(-3)^2?

  1. A−(3×3)-(3\times3)
  2. B(−3)×(−3)(-3)\times(-3)
  3. C(−3)×2(-3)\times2
Show answer and explanation
(−3)×(−3)(-3)\times(-3)

The parentheses put the minus sign inside the base, so both factors are −3.

03Check your understanding#

Check your reasoning

A tree starts with one branch. In each of 33 rounds, every branch is replaced by 22 new branches. How many branches are there after the final round?

Show answer and explanation
8

The counts are 2, then 4, then 8. This is 232^3.

Key takeaway

The exponent counts factors of the base.

  • Evaluate a positive whole-number power

Further questions

What does an exponent of one do?
It gives one copy of the base: 7 to the power 1 is 7.

Sources & further reading

  1. [1]

Reference this concept

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

Glacius. “Powers.” Math behind ML. /learn/f-powers