Square Roots Calculator

The square root of a number n is the value x such that x × x = n. So the square root of 25 is 5, because 5 × 5 = 25. Simple enough for perfect squares. For everything else, it gets a bit more involved.

For numbers that aren't perfect squares, you have a few options. You can use a calculator (the quickest route), apply an estimation method by finding the two perfect squares your number falls between, or use the long division method if you need a manual approach. Most people reach for a calculator, and honestly that's fine. But knowing the underlying logic helps you catch errors and build number sense.

The estimation approach works like this: say you want the square root of 50. You know that √49 = 7 and √64 = 8, so √50 is somewhere between 7 and 8, closer to 7. A calculator gives you 7.071. That kind of mental check is surprisingly useful.

The standard notation for a square root uses the radical symbol: √n. You can also write it as an exponent: n^(1/2) or n^0.5. Both forms mean exactly the same thing and show up regularly depending on the context, whether that's a textbook, a spreadsheet formula, or a programming language.

Formally, for any non-negative real number n:

• √n = x, where x ≥ 0 and x² = n
• The principal square root always refers to the positive root. For example, even though both 5 and -5 squared equal 25, √25 = 5 by convention.
• In exponential form: √n = n^(1/2)

This exponent form becomes especially handy in algebra and calculus, where radical expressions are often rewritten as fractional powers to make them easier to manipulate.

A perfect square is any integer that's the square of another integer. The numbers 1, 4, 9, 16, 25 are all perfect squares. Their square roots come out as clean whole numbers, with no decimals or radicals hanging around. That's what makes them so useful in math.

When you enter a perfect square into the calculator, the result is an exact integer. No rounding, no approximation. For everything else, you get a decimal that goes on indefinitely (an irrational number in most cases), so the calculator rounds it to a practical number of decimal places.

Recognizing perfect squares on sight saves real time on tests and in everyday calculations. The more familiar you are with them, the faster you can simplify expressions, reduce fractions, and spot patterns in number problems.

Simplifying a square root means rewriting it in its simplest radical form. The goal is to pull out any perfect square factors hiding inside the number under the radical sign. For instance, √72 can be simplified to 6√2, because 72 = 36 × 2 and √36 = 6.

This matters because simplified radical expressions are much easier to work with in equations, and most math teachers (and standardized tests) expect answers in simplified form. Leaving √72 as-is when it can be written as 6√2 is considered an incomplete answer.

The two most common approaches are the prime factorization method and a more intuitive factor-spotting approach. Both work fine. Prime factorization is more systematic and reliable for larger numbers.

Most square roots don't land on a whole number. The square root of 2, for example, is approximately 1.41421356... and those digits never repeat or end. It's irrational. For practical purposes, you round to however many decimal places you need.

Common decimal approximations worth knowing:

• √2 ≈ 1.414
• √3 ≈ 1.732
• √5 ≈ 2.236
• √6 ≈ 2.449
• √7 ≈ 2.646
• √8 ≈ 2.828
• √10 ≈ 3.162

For rough mental estimates, the method of bracketing between two known perfect squares works well. For anything requiring precision, use the calculator. In fields like engineering or construction, rounding errors can compound, so extra decimal places matter more than they might seem at first.

Working through a few examples covers a lot of ground quickly. Here's a range from straightforward to a bit more involved.

• √81: 9, because 9 × 9 = 81. Clean perfect square.
• √2: Approximately 1.414. Irrational, so we use a decimal approximation.
• √200: Simplify first. 200 = 4 × 50 = 4 × 25 × 2. So √200 = √(100 × 2) = 10√2 ≈ 14.142.
• √0.25: 0.5, because 0.5 × 0.5 = 0.25. Square roots of decimals work the same way.
• √(9/16): Take the square root of numerator and denominator separately. √9 / √16 = 3/4.
• √500: 500 = 100 × 5, so √500 = 10√5 ≈ 22.361.

Notice that simplifying before reaching for a decimal approximation often makes the work cleaner and the answer more useful in an equation context.

Here's where things get interesting. There's no real number whose square is negative, because any number times itself (positive or negative) produces a positive result. So √(-1) doesn't exist in the real number system.

That's where imaginary numbers come in. Mathematicians defined i as the square root of -1. It's called the imaginary unit. From there, the square root of any negative number can be expressed using i:

• √(-4) = 2i
• √(-9) = 3i
• √(-25) = 5i
• √(-7) ≈ 2.646i

In general: √(-n) = i√n for any positive number n.

Complex numbers, which combine a real part and an imaginary part (like 3 + 4i), rely on this concept. They're used heavily in electrical engineering, signal processing, and physics. For most everyday math, you won't run into negative square roots. But they're absolutely real in the mathematical sense, just in a different number system than the one we use by default.

This table covers the values that come up most often in algebra, geometry, and applied math. Bookmark it or keep it handy for reference.

The square root asks: what number multiplied by itself once gives me this value? The cube root asks: what number multiplied by itself twice (three times total) gives me this value? They're related concepts but serve different purposes.

One key difference: cube roots of negative numbers are real. Since (-2) × (-2) × (-2) = -8, ∛(-8) = -2 without any imaginary numbers involved. That's not the case for square roots, where negative inputs require i.

In practice, square roots come up far more often in basic math and geometry. Cube roots appear more in volume calculations and certain physics formulas. Both are special cases of the general nth root: n^(1/k).

Square roots aren't just an algebra exercise. They show up in real, practical situations more often than most people expect.

• Geometry: The Pythagorean theorem (a² + b² = c²) requires a square root to solve for any side of a right triangle. If you're calculating the diagonal of a room or the length of a ramp, you're using square roots.
• Area calculations: If you know the area of a square and need the side length, take the square root. A square with 144 square feet of area has sides of 12 feet.
• Physics: Formulas for velocity, energy, and wave frequency regularly involve square roots. The period of a pendulum, for example, depends on the square root of its length.
• Finance: Volatility in investment analysis is often expressed as a standard deviation, which is computed using a square root.
• Computer graphics: Distance between two points in 2D or 3D space uses the square root (via the distance formula, which extends the Pythagorean theorem).
• Construction and carpentry: The 3-4-5 right triangle rule used to check square corners relies directly on the Pythagorean theorem.

Pretty much anywhere you're dealing with areas, distances, or rates of change, square roots are somewhere in the math. Getting comfortable with them makes you more capable across a surprisingly wide range of fields.