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Square roots

The principal real square root of a nonnegative number is the nonnegative number whose square equals it.

1 min read1 figureSources & further reading
On this page 8 sections
  1. Overview
  2. A square root recovers a nonnegative number from its square
  3. A real number squared cannot be negative
  4. Check your understanding
  5. Key takeaway
  6. Further questions
  7. Sources & further reading
  8. Concept connections

01A square root recovers a nonnegative number from its square#

A square root recovers a nonnegative number from its square. A square with area 1616 has side length 44, because 42=164^2=16. Write this as 16=4\sqrt{16}=4.

A square with side length 4 has area 16; recovering the side length requires the nonnegative square root of 16.A square with side length 4 has area 16; recovering the side length requires the nonnegative square root of 16.
Figure 1A square with side length 4 has area 16; recovering the side length requires the nonnegative square root of 16.
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The symbol 16\sqrt{16} chooses 44, even though both 424^2 and (−4)2(-4)^2 equal 1616. This chosen value is the principal square root. It is nonnegative; 0=0\sqrt{0}=0.

Check your reasoning

Calculate 81\sqrt{81}.

Show answer and explanation
9

Nine is nonnegative and 92=819^2=81.

02A real number squared cannot be negative#

A real number squared cannot be negative. Positive and negative inputs both square to nonnegative values. Therefore −9\sqrt{-9} has no real value. In this lesson, the quantity under the root must be at least zero.

Check your reasoning

Which expression has no real value?

  1. A0\sqrt{0}
  2. B25\sqrt{25}
  3. C−25\sqrt{-25}
Show answer and explanation
−25\sqrt{-25}

There is no real number whose square is −25.

03Check your understanding#

Check your reasoning

A square has area 6464 square units. What is its side length, in units?

Show answer and explanation
8

The nonnegative length whose square is 64 is 88.

Key takeaway

A real square root has a nonnegative input and returns a nonnegative value.

  • Evaluate a real square root with its domain restriction

Further questions

Why is the square root not plus or minus?
The radical symbol names one value, the nonnegative root. Solving an equation such as x squared equals 16 is a different task and gives two solutions.

Sources & further reading

  1. [1]

Reference this concept

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

Glacius. “Square roots.” Math behind ML. /learn/f-roots