Root Calculator

Root Calculator

Calculated Result
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Source: Mathematical Standard Rules

 

Free Root Calculator: Square, Cube & Nth Root Solver

You might be an algebra student trying to simplify a complex problem. You might be a homeowner measuring a room. Or, you might be working in finance. Regardless of the reason, you need to understand roots. While exponents repeat multiplication, roots do the opposite. For many, this reverse process is hard to grasp.

Roots can get confusing quickly. You might feel comfortable with simple squares, but cube roots and negative numbers are much harder. You are likely here for a quick answer. However, you might also need to understand the method behind the math.

You have found the right place. Below, our Root Calculator will solve your problem instantly. At My Online Calculators, we go beyond simple answers. We act as a radical solver online that also teaches you the logic. We explain the manual methods and the formulas you need to know.

What is the Root Calculator?

This tool helps you find the “principal n-th root” of a number. Basic calculators often only handle the square root. A true solver must handle any degree. This means you can reverse any power.

In math terms, this calculator solves for $r$ in this equation:

$$r^n = x$$

  • $x$ (The Radicand): The number you want to find the root of.
  • $n$ (The Index): How many times $r$ is multiplied by itself to get $x$.
  • $r$ (The Root): The answer.

How to Use This Tool

We made this tool simple and fast. Follow these steps:

  1. Enter the Degree ($n$):
    • For a Square Root, enter 2.
    • For a Cube Root, enter 3.
    • For an Nth Root (like the 5th or 12th), enter that number.
  2. Enter the Number ($x$): Type the value you want to solve. You can use whole numbers, decimals, or negative numbers.
  3. Calculate: Click the button. The tool will give you the precise answer instantly.

The Square Root Formula Explained

To understand the result, you need to know the formula. Roots are the inverse of exponents. The symbol looks like this:

$$\sqrt[n]{x} = r$$

This means $$r^n = x$$.

In advanced math, we often write this as a fraction. This is the nth root of x formula:

$$\sqrt[n]{x} = x^{\left(\frac{1}{n}\right)}$$

For example, the cube root of 27 is the same as $27^{1/3}$. Both equal 3. Why? Because $3 \times 3 \times 3 = 27$. This relationship is key to solving complex algebra.

Complete Guide to Roots and Radicals

To master math, you need to understand how radicals behave. This section explains the parts of a root, the relationship between roots and exponents, and how to simplify radicals.

1. The Parts of a Radical

A radical expression has three main parts.

The Symbol ($\sqrt{}$)

This check-mark symbol indicates a root operation. It comes from the Latin word for “root.” The line over the numbers groups them together.

The Radicand ($x$)

This is the number inside the symbol. In $\sqrt[3]{8}$, the number 8 is the radicand. It is the value we are breaking down.

The Index ($n$)

This small number sits in the “hook” of the symbol.

  • Invisible 2: If you see no number, the index is 2 (square root).
  • Specific Numbers: A 3 means a cube root. A 4 means a fourth root.

2. Roots and Exponents Relationship

Roots and exponents are partners. One undoes the other.

  • Subtraction undoes Addition.
  • Division undoes Multiplication.
  • Roots undo Exponents.

If you square 5 ($5^2$), you get 25. The square root of 25 ($\sqrt{25}$) brings you back to 5. You can always write a radical as a power using the fractional rule:

$$\sqrt[n]{a^m} = a^{\frac{m}{n}}$$

This rule helps you use the laws of exponents to solve hard equations.

3. How to Simplify Radicals

You can combine roots using specific laws. These help you simplify radicals without a calculator.

The Product Rule

You can multiply two roots if they have the same index.

$$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$$

Example: Simplify $\sqrt{18}$.
We know $18 = 9 \times 2$.
$\sqrt{18} = \sqrt{9} \times \sqrt{2}$.
Since $\sqrt{9} = 3$, the answer is $3\sqrt{2}$. This is the “exact form.”

The Quotient Rule

You can split a fraction inside a root.

$$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$

Example: $\sqrt{\frac{16}{25}}$ becomes $\frac{\sqrt{16}}{\sqrt{25}}$. This equals $\frac{4}{5}$ or 0.8.

4. How to Calculate Roots Manually

Students often ask how to calculate roots manually. There are two main ways to do this without a phone.

Method A: Prime Factorization

This is best for exact answers.

  1. Break it down: Find the prime factors of the number. For $\sqrt[3]{216}$, the factors are $2 \times 2 \times 2 \times 3 \times 3 \times 3$.
  2. Group them: Since the index is 3, group them in threes. We have a group of 2s and a group of 3s.
  3. Extract: Pull one number from each group. $2 \times 3 = 6$. The answer is 6.

Method B: Estimation

What if the number isn’t perfect, like $\sqrt{10}$?

  1. Guess: 10 is close to 9. Since $\sqrt{9}=3$, the answer is roughly 3.
  2. Divide: Divide 10 by 3. You get 3.33.
  3. Average: Average your guess (3) and the result (3.33). $(3+3.33)/2 = 3.165$.
  4. The real answer is 3.162, so this simple math got us very close.

5. The Root of a Negative Number

Can you find the root of a negative number? It depends on the index.

  • Odd Index (3, 5, 7): Yes. You can multiply a negative number by itself an odd number of times to get a negative result. $\sqrt[3]{-8} = -2$.
  • Even Index (2, 4, 6): Not in the real world. No real number squared equals a negative. Mathematicians use “Imaginary Numbers” (like $i$) to solve these.

Real-World Applications

Roots define geometry and finance. Here is why they matter.

  • Finance: Investors use roots to calculate growth. The Compound Annual Growth Rate (CAGR) uses an $n$-th root to show how an investment grows over time.
  • Construction: Builders use the square root to find diagonal distances. This ensures corners are perfectly square.
  • Electricity: Engineers calculate RMS (Root Mean Square) voltage. This helps measure the power of alternating current in your home.

Frequently Asked Questions (FAQ)

1. How do I simplify radicals with variables?

Use the division rule. Divide the exponent inside by the index outside. For $\sqrt{x^5}$, divide 5 by 2. You get 2 with a remainder of 1. Pull $x^2$ out, and leave $x^1$ inside. The answer is $x^2\sqrt{x}$.

2. Is there a cube root calculator with steps?

Our tool provides the instant numerical answer. To show steps for homework, use the Prime Factorization method explained above. Break the number into prime parts and group them by three.

3. Why does the square root have two answers?

Technically, $5 \times 5 = 25$ and $-5 \times -5 = 25$. So, 25 has two roots: 5 and -5. However, the $\sqrt{}$ symbol usually asks for the “Principal Root,” which is the positive one.

4. What is the nth root of x?

This is the general term for any root. “N” stands for the number of the root (the index), and “X” is the number you are solving. It represents the formula $\sqrt[n]{x}$.

Ready to solve? Scroll up to the top and use the Free Root Calculator now.

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People also ask

A root calculator finds the nth root of a number. That means it returns the value that, when raised to the power n, gives the original number.

Example: √25 = 5, because 5^2 = 25.

They’re all roots, the only change is the value of n:

  • Square root: n = 2 (written as √x)
  • Cube root: n = 3 (written as ³√x)
  • Nth root: any whole number n (written as ⁿ√x)

Example: ⁴√16 = 2, because 2^4 = 16.

It depends on the number:

  • If the input is a perfect power (like 25 for square roots, or 27 for cube roots), the result can be exact.
  • If not, the calculator gives a decimal approximation, rounded to a certain number of digits.

Example: √2 ≈ 1.41421356..., so most calculators show a rounded version.

Some can, and some can’t. A simplification-focused tool may rewrite radicals by factoring out perfect squares (or perfect cubes, etc.).

Example: √50 can be simplified to 5√2 because 50 = 25 × 2.

If your calculator only shows decimals, you might see √50 ≈ 7.071... instead.

It depends on whether the root is even or odd:

  • Even roots (like square roots) of negative numbers aren’t real numbers. Many calculators will show an error, “no real root,” or a complex (imaginary) result.
  • Odd roots (like cube roots) of negative numbers are real.

Example: ³√(-8) = -2, because (-2)^3 = -8.

Most show the principal root, which is the nonnegative root for square roots.

So the calculator’s value is often correct, but your math problem may require both signs.

Many calculators use fast numerical methods that improve a guess again and again until it reaches the requested precision. A common approach is an iterative method (often taught as Newton’s method in algebra and calculus).

In plain terms, it keeps correcting the estimate until the digits stop changing.

Most are accurate to the digits they show, as long as you understand that many roots are rounded. Accuracy also depends on the tool’s precision setting (how many decimals it calculates and displays).

A quick check you can do: take the calculator’s result and raise it to the nth power. It should match the original number closely, based on the rounding.

Many do, especially student-focused tools. Step-by-step versions may show methods like:

  • factoring to simplify radicals
  • an iterative process for decimal answers

If you need to submit work, choose a calculator that explains the process, not just the final number.

Often yes, but limits vary by tool. Some calculators cap decimal places to stay fast, and some may struggle with very large inputs.

If you need lots of digits (or exact radical form), look for a tool that supports precision settings or symbolic simplification, not just a basic decimal output.