Square Root Calculator

This Square Root Calculator finds √x in real time, with the principal root, the negative root, a simplified radical when one exists, and a square mode for the reverse check.

Stuck between memorized perfect squares and a decimal that won't settle? You're not alone.

Read on and you'll also find out how to simplify √72 the way teachers expect, estimate √70 by hand, and read a coefficient like 2√5 without mixing it up with √10.

Here's what you'll walk away knowing:

• How to run the Square Root Calculator and read ± answers
• What a square root really is, in plain language
• How to simplify radicals like √72 → 6√2 by hand
• Mental estimates, fast calculator methods, and the 2√5 notation trap

How to Use the Square Root Calculator

Four quick moves. Results update as you type.

1. 1

   Pick Square root or Square (x²)

   Square root mode solves for √x. Square mode locks in x² when you need the reverse path.

2. 2

   Enter the number (the radicand when you are finding a root)

   Try positives like 72, decimals like 9.81, or a negative if you want the imaginary answer.

3. 3

   Set decimal places if you want tighter rounding

   Default precision is fine for homework. Bump it when you’re matching a printed key.

4. 4

   Read principal, negative, simplified, and check rows

   Note that the principal (non-negative) square root is what most calculators return for √x when x ≥ 0. The second answer is −√x. Use the calculator for speed. Learn the steps so you can check the answer.

What Is a Square Root, Really?

Picture Sarah staring at a square patio sketch with area 72 square feet. She doesn't want the area again. She wants one side length. That side is the square root. Geometry named it.

A square root is a number that multiplies by itself to give the original value. For a non-negative number x, the principal square root √x is the non-negative answer y where y² = x. The radicand sits under the radical sign. Positive x also has a matching negative root −√x, because (−y)² equals y² too.

Imagine Sarah is checking √72 next to nearby perfect squares. She knows 8² = 64 and 9² = 81. So √72 sits between 8 and 9. Same story for √70: closer to 8 than to 9, roughly 8.37. Pretty easy, isn't it?

How do you solve √ on a worksheet when the teacher wants both forms? Start with perfect-square factors, write the simplified radical, then peek at a decimal only if the problem asks for it. For √32 you get 4√2 ≈ 5.657. For √75 you get 5√3 ≈ 8.660. For √12 you get 2√3 ≈ 3.464. Sound familiar if you've been drowning in decimal-only answer keys?

So what does 2√5 mean? It means 2 × √5, not √(2 × 5). Mix those up and your homework quietly drifts. See the difference when 2√5 ≈ 4.472 while √10 ≈ 3.162?

Is 72 a perfect square? No. √36 is the clean integer case people memorize first. √27 simplifies to 3√3, which is not an integer either. Perfect squares like 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 give integer roots. Most other radicands stay irrational after you simplify.

Apart from this, √(a²) is |a|, not always a. Plug in −5 and you still get 5 from the principal root of 25. You cannot take a real square root of a negative number. Complex answers use i, like √(−4) = ±2i. This won't apply if your course bans imaginary numbers, so stay in the positives for that class.

Now you know why we print two answers plus a simplified form. That lived-in habit among tutors: we still flash the perfect squares from 1² through 15² before guessing anything messy, because your eye lands faster than your fingers.

The Square Root Formula

Here's the relationship that runs everything:

• x is the radicand, the number under the root.
• √x is the principal (non-negative) square root when x ≥ 0.
• −√x is the other real solution of y² = x.
• Simplifying means factoring out perfect squares so the leftover radicand is as small as you can get.

Sarah's √72, by hand

Now let's plug in her value.

1. Factor: 72 = 36 × 2, and 36 is a perfect square.
2. Split: √72 = √(36 × 2) = √36 × √2.
3. Pull out: √36 = 6, so √72 = 6√2.
4. Decimal check: √2 ≈ 1.4142, so 6 × 1.4142 ≈ 8.485.

So that gives us both the clean radical form and a decimal she can compare with the tool. For 3√72, she'd simplify inside first: 3 × 6√2 = 18√2. Here's where it gets interesting for exponents: because √x = x^1/2, raising a square root to the second power returns the radicand (for x ≥ 0). That identity is how you check work without trusting a screen.

Let's also peek at √(a/b) when both pieces are perfect squares. √(4/9) = 2/3. Fractions play fair once both tops and bottoms simplify. Typically you still rationalize a leftover root in the denominator if your teacher wants that style.

Of course, you can skip all this counting and let the Square Root Calculator do it instantly.

Where Square Roots Show Up Outside Class

Roots aren't only worksheet ornaments. They sneak into measurements, money math, and mental shortcuts.

Priya prices a square tile floor of 72 ft². She needs one side length, not a lecture. √72 ≈ 8.485 ft. Units don't invent a new formula here. Taking √ of inches still lands in length units for a square layout, but "√72 inches" isn't a special method. It's the same radicand with a unit label attached.

Consider Kevin on a quiz with no calculator. He wants √70. He brackets it: 8² = 64, 9² = 81. Seventy sits nearer 64, so he tries 8.4² = 70.56, then 8.3² = 68.89. Midway near 8.37 usually works for a fast estimate. How can you quickly calculate square roots in your head for a three-digit number? Same sandwich. For √144 you already know 12. For √150, sit between 12² = 144 and 13² = 169, then nudge toward 12.25. Any-digit estimation is just better brackets plus one careful square check.

What about Maria comparing investment volatility when a formula coughs up a variance of 25? The standard deviation is √25 = 5. Finance loves roots that way. Her sheet still needs the principal root, not −5, unless a model says otherwise.

And Daniel racing through perfect squares remembers the 25 × 25 pattern for squaring numbers that end in 5: 25² = 625, because you take the leading digit times one more (2 × 3 = 6) and glue on 25. Handy when you're checking roots of near squares by hand. (good luck with that mid-test!)

Besides homework drills, right triangles keep showing up. A ladder propped 9 ft from a wall with a 15 ft hypotenuse leaves height √(15² − 9²) = √144 = 12 ft. That's Pythagorean work that quietly starts with square roots. Link out to the Pythagorean Theorem Calculator when the right triangle is the real puzzle and the root is just a step.

Common Mistake: Reading 2√5 as √10

Here's where most people get confused about radicals. A coefficient sitting in front looks like it should slide under the root. It doesn't.

2√5 means two times the square root of five. √(2 × 5) would be √10. Those are different numbers. Another classic slip: treating √72 as "about 9" because 81 is nearby, without checking that 8² = 64 is closer. Why does that matter here? Estimation errors stack when you also skip simplifying.

But √(a²) is |a|. And negative radicands don't give real answers. So don't force a real√ on −9 in a course that only grades reals.

That's why we show simplified form next to decimals. Cross-check both, then move on.

FAQs

Need full equation solving with roots on both sides? Keep it brief here and jump to the Quadratic Formula Calculator. For cube roots or higher, use the Root Calculator.

References

1. Weisstein, Eric W. "Square Root." MathWorld — A Wolfram Web Resource. https://mathworld.wolfram.com/SquareRoot.html
2. OpenStax. "1.3 Radicals and Rational Exponents." College Algebra 2e. OpenStax / Rice University. https://openstax.org/books/college-algebra-2e/pages/1-3-radicals-and-rational-exponents
3. OpenStax. "8.2 Simplify Radical Expressions." Intermediate Algebra 2e. OpenStax / Rice University. https://openstax.org/books/intermediate-algebra-2e/pages/8-2-simplify-radical-expressions
4. OpenStax. "8.1 Simplify Expressions with Roots." Intermediate Algebra. OpenStax / Rice University. https://openstax.org/books/intermediate-algebra/pages/8-1-simplify-expressions-with-roots
5. OpenStax. "9.2 Simplify Square Roots." Elementary Algebra 2e. OpenStax / Rice University. https://openstax.org/books/elementary-algebra-2e/pages/9-2-simplify-square-roots
6. OpenStax. "3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem." Elementary Algebra 2e. OpenStax / Rice University. https://openstax.org/books/elementary-algebra-2e/pages/3-4-solve-geometry-applications-triangles-rectangles-and-the-pythagorean-theorem
7. NIST / SEMATECH. "1.3.5.6. Measures of Scale." e-Handbook of Statistical Methods. National Institute of Standards and Technology. https://www.itl.nist.gov/div898/handbook/eda/section3/eda356.htm
8. Olver, F. W. J., et al. (eds.). "§4.2 Definitions" (powers / principal values). NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/4.2

Note: Floating-point square-root hardware and library algorithms vary by platform. The page's Newton / Babylonian description is a teaching model for the common divide-and-average iteration, not a claim about every CPU or language runtime.