Square Root Calculator

All About Square Root Calculator

Square roots can seem mysterious. That little √ symbol shows up in math problems like it expects us to just get it. But really, square roots are just the opposite of squaring a number, and they show up everywhere — in geometry, algebra, physics, even finance. The Symbolab Square Root Calculator helps make sense of it all by simplifying, solving, and showing you every step along the way.

What Is a Square?

Before we talk about square roots, it helps to understand what a square is in math. To square a number means to multiply it by itself.

$4^2 = 4 \times 4 = 16$

In words: four squared is sixteen.

We call it “squaring” because if you made a square with sides of length 4 units, the area inside that square would be 16 square units. This connection between numbers and geometry is where the name comes from.

Here are a few more examples:

$3^2 = 9$,
$10^2 = 100$,
$(-4)^2 = 16$

Even negative numbers become positive when squared because a negative times a negative equals a positive.

Why Does a Negative Times a Negative Equal a Positive?

This is a really common question, and a good one. Think about multiplication as movement on a number line. A positive number pushes you to the right.

A negative number flips your direction.

So if you do a negative times a negative, it’s like flipping directions twice — which brings you back to the original direction: positive. Still confused? Try this: Imagine you're reversing your bike 3 miles backwards every day for 2 days.

That’s: $-3 \times 2 = -6$

You went 6 miles in reverse that makes sense. Now imagine we go back in time 2 days. If you ask, "Where was the bike before it went backwards 3 miles per day?" the math flips again: $-3 \times -2 = 6$

We’re reversing a reverse, and we end up moving forward.

That’s why: $(-5)^2 = (-5) \times (-5) = 25$

Two negatives make a positive when you multiply them.

What Is a Square Root?

If squaring a number means multiplying it by itself, then taking the square root means asking the reverse question:

“What number was multiplied by itself to get this?”

It’s like playing math detective, working backwards to figure out what caused the result.

For example:

$49 = 7^2$ because $7 \times 7 = 49$

That √ symbol is called a radical, and the number inside it is the radicand. When you take the square root of a number, you're looking for the value that was squared to get it.

Now here’s something important: the square root symbol always gives you the principal square root, which means the positive one.

Yes, both 6 and –6 squared give 36:

$(-6)^2 = (-6) \times (-6) = 36$

But: $\sqrt{36} = 6$

We only write the positive root when we use the radical symbol, so $\sqrt{36} = 6$.
But if solving an equation like $x^2 = 36$, the solutions are $x = 6$ and $x = -6$.

Can We Take the Square Root of Any Number?

Well, almost.

• If the number is positive, you can take the square root and get a real number.-
• If the number is not a perfect square, the answer will be a decimal or an irrational number (more on that later).-
• If the number is negative, you can still take the square root, but the answer will involve imaginary numbers.

A Quick Look at Imaginary Numbers

So what happens when you try to take the square root of a negative number? Let’s say you’re staring at:

$\sqrt{-9} = 3i$

That might seem impossible at first. And you’re right — if we stick to real numbers, there’s no solution. There’s no number that you can multiply by itself to get a negative.

That might seem impossible at first. And you’re right — if we stick to real numbers, there’s no solution. There’s no number that you can multiply by itself to get a negative.

Why? Because:

• A positive times a positive is positive
• A negative times a negative is also positive
• So we never get a negative by squaring a real number.

To solve this, mathematicians invented something called an imaginary number, using the symbol i, which is defined as:

$i = \sqrt{-1}$

This lets us write answers like:

$$\sqrt{-9} = \sqrt{9 \times (-1)} = \sqrt{9} \cdot \sqrt{-1} = 3i$$

Real-world example:

Imagine your school is keeping track of daily temperature changes throughout the week, how much warmer or colder each day is compared to the one before.

Let’s say the math model they’re using to track patterns ends up with a square root of a negative number. It doesn’t mean the temperature is really imaginary. It just means the pattern is flipping direction, maybe even looping back in time, like a reset.

In more advanced math, imaginary numbers help model changes that move back and forth, like waves, cycles, or signals—anything that does not just move in a straight line.

So while you might not use $i$ to measure your lunch break, it is helping people behind the scenes track movement, rhythm, and flow in everything from climate patterns to music apps.

Types of Square Roots

Some square roots are neat and perfect. Others are a little messier. And a few steps into the world of imaginary numbers. Understanding these types will help you know what kind of answer to expect and how to simplify it when possible.

1. Perfect Square Roots

These are the ones we like best when we’re first learning. A perfect square is a number that comes from squaring a whole number. So its square root is also a whole number.

• $\sqrt{36} = 6$
• $\sqrt{225} = 15$
• $\sqrt{100} = 10$

These are the easiest to work with, and you’ve probably memorized a few already. They show up often in geometry and algebra problems.

2. Irrational Square Roots

But not every number is that tidy. Some numbers, like 2, 3, or 5, are not perfect squares. Their square roots go on forever without repeating, and we can’t write them exactly as fractions. These are called irrational numbers.

For example:

$\sqrt{2} \approx 1.4142\ldots$

It’s a decimal that never ends and never settles into a pattern. That’s what makes it irrational. These kinds of square roots come up when you’re solving problems with diagonals, circles, or right triangles. You’ll often leave them in square root form, like $\sqrt{5}$, unless you need a decimal estimate.

3. Square Roots of Fractions

You can take the square root of a fraction too, and it’s not as scary as it sounds. As long as both the top and bottom are perfect squares, you can take the square root of each part separately:

$\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}$

Even if they aren’t perfect squares, the calculator can still simplify or approximate them. This is useful when you’re working with proportions, percentages, or probabilities — situations where fractions are everywhere.

4. Square Roots of Decimals

You can also take square roots of decimals. If the decimal is a perfect square, you’ll get a nice result:

$\sqrt{0.25} = 0.5$

If not, you’ll get an irrational number again, and a calculator will be your best friend. These come up in science, measurements, or any real-life situation where values aren’t whole numbers.

5. Square Roots of Negative Numbers

This is where we revisit the world of imaginary numbers.

You cannot take the square root of a negative number and stay in the real number system,
but you can write it using $i$, the imaginary unit.

For example:

$\sqrt{-16} = 4i$

This kind of root shows up in more advanced math — especially when solving certain equations, working with complex numbers, or modeling wave-like behavior.

No matter which type of square root you’re working with — perfect, irrational, fractional, decimal, or imaginary, Symbolab’s Square Root Calculator can handle it. It knows when to simplify, when to estimate, and when to switch into imaginary mode. And it explains the steps along the way, so you’re never just guessing.

How to Simplify Square Roots (Manually)

Sometimes square roots give us a clean answer.

But other times, we’re left with something like $\sqrt{72}$ and we need to simplify it. That means rewriting it in a way that pulls out any perfect squares hiding inside.

Here’s how you do it, step by step.

Step 1: Find a perfect square factor

Look at the number inside the square root, the radicand, and ask yourself, “Does this number have any perfect square factors?”

Take $\sqrt{72}$ ​ for example.

72 isn’t a perfect square, but it can be factored into: 72=36×2

And 36 is a perfect square.

Step 2: Break it up

Now you break the square root into two parts:

$\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2}$

Step 3: Simplify the perfect square

$\sqrt{36} = 6$

So,

$\sqrt{72} = 6\sqrt{2}$

Let’s try another: $\sqrt{50}$

$50 = 25 \times 2$

$\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}$

Even if the radicand looks random at first, there is often a perfect square tucked inside just waiting to be pulled out.

What About Rationalizing the Denominator?

Sometimes you will see a square root in the denominator of a fraction, like:

$\frac{1}{\sqrt{2}}$

In most cases, we want to get rid of the square root in the denominator.
This is called rationalizing the denominator.

To do that, multiply the top and bottom by the same square root:

$\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

Now the denominator is rational, and the expression is simplified.

Why Do We Still Learn to Do This by Hand?

You might be thinking, “Why go through all this when I can just use a calculator?” Fair question. The truth is, learning to simplify square roots manually:

• Helps you understand what’s happening behind the scenes.
• Gets you ready for algebra and geometry problems.
• Makes you better at spotting patterns in numbers.
• Builds confidence in your math skills.

That said, manual methods can be time-consuming and easy to mess up, especially with bigger numbers. That’s why tools like Symbolab’s Square Root Calculator are so helpful. They do the heavy lifting and show every step, so you can check your work or learn the process more deeply.

What Is a Square Root Calculator?

A square root calculator is more than just a fancy number cruncher. It’s a math helper that takes the stress out of simplifying or solving square roots, especially when the numbers get big or messy.

When you enter something like $\sqrt{72}$ into Symbolab’s Square Root Calculator, it does three important things:

• It simplifies the radical, if possible.
• It gives a decimal approximation.
• It shows you how it got the answer, step by step.

How Symbolab’s Square Root Calculator Works

Using Symbolab’s Square Root Calculator feels less like using a machine and more like working with a math-savvy friend who always shows their work. Here’s how it works and what makes it so powerful.

Step 1: Enter the Problem

To get started, all you have to do is enter your problem in the way that works best for you:

• Type it in using your keyboard or Symbolab’s on-screen math pad.
• Upload a photo of a handwritten problem (yes, even messy notes from your notebook).
• Paste a screenshot from a digital worksheet or photo of a handwritten equation.

You don’t need to perfect formatting — Symbolab can read your intent, parse the math correctly, and get to work. Once you're ready, click the big red Go button. Symbolab’s Square Root Calculator fast, intuitive, and student-friendly.

Step 2: Your Personalized Solution Page

After you click Go, you’re taken to a full solution page. Here's what you'll find:

• Simplified Result: The square root is simplified right away. If the answer involves a radical and a decimal, you’ll see both.
• Decimal Approximation: Perfect for when you need a real-world number — say, for measuring length or comparing values in a graph.
• Step-by-Step Box: This is where Symbolab really shines. It breaks down each part of the problem and shows you how the final answer was built.

And if you want to slow it down?

• Toggle "One step at a time" to reveal the solution gradually.
• Use the hint feature if you’re stuck and need a nudge (without spoiling the whole answer).
• Hover over or tap on math rules and symbols to get quick refreshers.
• Use the “Chat with Symbo” help tool to ask a question or get clarification instantly.

Step 3: Built-in Support Features

Symbolab isn’t just about the math. It’s about how you learn the math. That’s why the calculator includes tools to help you study smarter:

• Highlighting and collapsing steps so you can focus on just what you need.
• Clear labels and explanations for each transformation, like exponent rules or radical properties.
• Real-time interactivity, not just a static answer.
• Mobile- and browser-friendly design, so you can use it on a phone, tablet, or a computer. Now it also has a Chrome extension.
• Error detection and correction if you enter something a little off, the calculator often suggests fixes or catches missing parentheses.

Whether you're working through practice problems, checking homework, or prepping for a quiz, Symbolab’s Square Root Calculator acts like a digital math coach. It does more than solve — it teaches, supports, and helps you build confidence along the way.

Quick Recap:

• Type, snap, or upload your square root problem.
• Get an answer instantly, both exact and decimal.
• Learn the logic with step-by-step explanations.
• Get support through hints, chat, and built-in explanations.

It’s built not just for speed, but for learning. Use it to double-check homework, review before tests, or figure out where you got stuck in class.

Applications of Square Roots in Real Life

Square roots help us solve problems in the real world, from building things to analyzing data.

• Geometry: Find the diagonal of a square or right triangle using the Pythagorean Theorem.-
• Physics: Calculate speed, energy, or motion in real-world science problems.-
• Finance: Measure risk and variability using standard deviation.-
• Engineering: Solve for stress, force, or material strength in design work.-
• Technology & Games: Track distance and motion in 3D animation and video game physics.

Common Mistakes When Working with Square Roots

These are the slip-ups students make most often — easy to miss, but easy to fix once you know what to look for.

• Forgetting ± in equations: Solving $x^2 = 25$ gives two answers: $x = 5$ and $x = -5$, not just one.

• Misusing the square root symbol: $\sqrt{25}$ means only the positive root, not $\pm 5$.

• Breaking up square roots incorrectly: $\sqrt{a + b} \ne \sqrt{a} + \sqrt{b}$

• Leaving radicals in the denominator: Always rationalize expressions like $\frac{1}{\sqrt{3}}$.

• Assuming negative square roots are undefined: $\sqrt{-9} = 3i$, not “no solution.”

• Rounding too early: Keep square roots exact (e.g., $\sqrt{2}$) until the final step to avoid rounding errors.

• Missing perfect square factors: Simplify $\sqrt{50} = 5\sqrt{2}$, do not just leave it as-is.

• Calculator input errors: Misplaced parentheses or missing square root signs can change the entire answer.

Square roots are everywhere, in math problems, real-life decisions, and the way we understand patterns and space. The Symbolab Square Root Calculator helps make sense of them, offering instant answers and step-by-step explanations that build real understanding. Whether you're simplifying by hand or checking your work, the calculator is a powerful tool for learning. Use it to explore, practice, and grow more confident with every problem. Math gets easier when you have the right support and Symbolab Square Root Calculator is a great place to start.