Square Roots: Definition with Solved Problems

The square root is one of the main topics in Mathematics. This page includes simplifications of many square roots in their simplest radical forms.

Definition of Square Root

If two numbers x and d satisfy the following equation

x2 = d,

then the number x is called a square root of the number d. Symbolically, we write it as follows


For example, we have 72 = 49, so from the above definition of square roots we can conclude that 7 is a square root of 49.

Properties of Square Roots

Square roots have the following properties:

  • The square root of zero is always zero.
  • A square root of a number can be positive or negative. For example, 5 and -5 are square roots of 25.
  • Two square roots can be multiplied as follows √a × √b = √(ab).
  • Division rule for square roots: √a ÷ √b = √(a/b) where b is non-zero.
  • Sum/addition rule of square roots: √a + √b ≠ √(a+b) may not be TRUE.
How to Multiply Square Roots: 4 Ways

Simplifying Square Roots

Now, we will provide a step-by-step discussion on how to simplify a square root of a number in its simplest radical form. For example,

Root 8 in radical form: √8 = 2√2.

Square root 12 simplified: √12 = 2√3.

How to simplify root 18: √18 = 2√3.

How to simplify root 20: √20 simplified is 2√5.

How to simplify root 27 in simplest radical form: √27 = 3√3.

Root 50 simplified: √50 = 5√2.

Square Root of Negative Numbers

Square root of -4: √-4 =2i.

Square root of -9: √-9 = 3i.

Fourth Root:

Fourth root of 625 is 5

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