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Determine square roots of perfect squares

A square root of a nonnegative perfect square is the nonnegative number that, when multiplied by itself, produces the given number; for example, √144 = 12 because 12² = 144. The understanding includes identifying perfect squares, including 0, and distinguishing the principal square root, √a, from the two solutions of x² = a, namely x = ±√a when a is positive. The focus is on exact roots of familiar whole-number perfect squares, not approximating irrational roots or developing general radical simplification.

Detailed Explanation: Determine square roots of perfect squares

A perfect square is a whole number that can be written as a whole number multiplied by itself. For example,

122=12⋅12=14412^2=12\cdot 12=144

So, the principal square root of (144)(144) is (12)(12).

Example: Find 144\sqrt{144}

  1. Ask: What number multiplied by itself equals (144)(144)?

  2. Check (12)(12):

12⋅12=144 12\cdot 12=144

  1. Therefore,

144=12 \boxed{\sqrt{144}=12}

The symbol 144\sqrt{144} means the principal square root, which is the nonnegative answer. It is not (−12)(-12).

However, if the question is to solve an equation such as

x2=144,x^2=144,

then both (12)(12) and (−12)(-12) work:

122=144and(−12)2=144.12^2=144 \qquad \text{and} \qquad (-12)^2=144.

Thus, the solutions are

x=±12.\boxed{x=\pm 12}.

Remember that 00 is also a perfect square because

02=0,0^2=0,

so 0=0\sqrt{0}=0.

Learn by doing: Determine square roots of perfect squares

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    Square Roots of Perfect Squares

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