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Principal Square Root — Definition, Formula & Examples

The principal square root of a number is its non-negative square root. When you see the radical symbol √, it always refers to the principal (positive) square root.

For any non-negative real number aa, the principal square root, denoted a\sqrt{a}, is the unique non-negative number bb such that b2=ab^2 = a.

Key Formula

a=bwhere b≥0 and b2=a\sqrt{a} = b \quad \text{where } b \geq 0 \text{ and } b^2 = a
Where:
  • aa = A non-negative real number (the radicand)
  • bb = The principal (non-negative) square root of a

How It Works

Every positive number actually has two square roots: one positive and one negative. For example, both 55 and −5-5 square to 2525. The radical symbol x\sqrt{\phantom{x}} is defined to return only the non-negative root, which is called the principal square root. If you want the negative root, you write −a-\sqrt{a}. To indicate both roots at once, you write ±a\pm\sqrt{a}.

Worked Example

Problem: Find the principal square root of 36.
Identify both square roots: Both 6 and −6 satisfy the equation x2=36x^2 = 36.
62=36and(−6)2=366^2 = 36 \quad \text{and} \quad (-6)^2 = 36
Choose the non-negative value: The principal square root is the non-negative one.
36=6\sqrt{36} = 6
Answer: 36=6\sqrt{36} = 6, not −6-6.

Why It Matters

When you solve x2=49x^2 = 49 in algebra, the answer is x=±7x = \pm 7, but when you simplify 49\sqrt{49}, the answer is just 77. Understanding this distinction prevents errors in solving quadratic equations and simplifying radical expressions throughout algebra and beyond.

Common Mistakes

Mistake: Writing 25=±5\sqrt{25} = \pm 5.
Correction: The radical symbol x\sqrt{\phantom{x}} returns only the non-negative value, so 25=5\sqrt{25} = 5. You only write ±\pm when solving an equation like x2=25x^2 = 25, which gives x=±5x = \pm 5.