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Square Root

Square Root

A nonnegative number that must be multiplied times itself to equal a given number. The square root of x is written Square root symbol: √x or x½. For example, Square root of 9 equals 3 since 32 = 9.

Note: Square root symbol √x, representing the nonnegative square root of x never refers to a negative number. Even though (–3)(–3) = 9, we do not say that –3 is a value of Square root of 5, written as √5. Also, if x itself is negative then Square root symbol √x, representing the nonnegative square root of x is imaginary.

 

 

See also

nth root, cube root, square root rules, radical rules, rational exponents

Key Formula

x=ymeansy2=xwhere y≥0\sqrt{x} = y \quad \text{means} \quad y^2 = x \quad \text{where } y \geq 0
Where:
  • xx = The nonnegative number under the radical sign (the radicand)
  • yy = The nonnegative number whose square equals x

Worked Example

Problem: Simplify √72.
Step 1: Find the largest perfect square that divides 72. Since 36 × 2 = 72, and 36 is a perfect square, use 36.
72=36×272 = 36 \times 2
Step 2: Apply the product rule for square roots: the square root of a product equals the product of the square roots.
72=36×2=36⋅2\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2}
Step 3: Evaluate √36, which is 6, and write the simplified form.
36⋅2=62\sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}
Answer: √72 = 6√2, which is approximately 8.485.

Another Example

Problem: Solve for x: √x = 7.
Step 1: To undo the square root, square both sides of the equation.
(x)2=72\left(\sqrt{x}\right)^2 = 7^2
Step 2: Simplify both sides. Squaring a square root returns the original value.
x=49x = 49
Step 3: Check: √49 = 7 ✓. The solution is valid.
49=7\sqrt{49} = 7
Answer: x = 49.

Frequently Asked Questions

Can a square root be negative?
The principal square root, written with the radical symbol √, is always nonnegative by definition. While both 3 and −3 satisfy x² = 9, the expression √9 equals only 3, not −3. If an equation requires both solutions, you write ±√9 = ±3.
What is the square root of a negative number?
The square root of a negative number is not a real number — it is an imaginary number. For example, √(−4) = 2i, where i is the imaginary unit defined by i² = −1. You typically encounter these in algebra II and beyond.

Square Root vs. Cube Root

A square root asks "what number times itself equals x?" while a cube root asks "what number times itself three times equals x?" The square root of 64 is 8 (since 8² = 64), but the cube root of 64 is 4 (since 4³ = 64). A key difference: cube roots can be taken of negative numbers and produce real results (∛(−8) = −2), whereas square roots of negative numbers are imaginary.

Why It Matters

Square roots appear constantly in geometry — the Pythagorean theorem, distance formulas, and area calculations all rely on them. They are also central to solving quadratic equations using the quadratic formula. In science and statistics, square roots show up in formulas for standard deviation, wave behavior, and gravitational calculations.

Common Mistakes

Mistake: Assuming √(a² + b²) equals a + b.
Correction: Square roots do not distribute over addition. For example, √(9 + 16) = √25 = 5, not 3 + 4 = 7. You must add under the radical first, then take the root.
Mistake: Forgetting that √x² = |x|, not simply x.
Correction: Since the square root always returns a nonnegative value, √x² = |x|. For instance, if x = −5, then √((−5)²) = √25 = 5 = |−5|, not −5.

Related Terms