Radical Calculator - Nth Root Finder

This nth root calculator helps you find any root of a number—whether that's a familiar square root, a cube root, or something less common like a 4th or 5th root. Just enter your number and the root degree, and you'll get your answer instantly.

Whether you're double-checking homework, working through a problem for class, or solving a calculation that came up at work, this tool makes finding roots straightforward and fast. No need to remember the formula or dig out a scientific calculator—just plug in your numbers and go.

An nth root is essentially the reverse of raising a number to a power. When you calculate the 4th root of 16, you're asking: "What number, multiplied by itself 4 times, equals 16?" The answer is 2, because 2 × 2 × 2 × 2 = 16.

The "n" in nth root simply refers to how many times that root number needs to multiply by itself. You're probably already familiar with the most common roots:

• Square root (2nd root): What times itself equals this number?
• Cube root (3rd root): What times itself three times equals this number?
• 4th root, 5th root, and beyond: Same idea, just more multiplications

The concept stays the same no matter how high the root goes. A 7th root asks what number, multiplied by itself 7 times, gives you the original number.

Now that you understand what an nth root actually is, let's look at the math behind it.

The formula for calculating any nth root is more approachable than it looks:

ⁿ√x = x^(1/n)

In everyday terms: to find the nth root of a number, raise that number to the power of 1 divided by n.

Worked Example:

Let's find the 4th root of 67 (the example shown in our calculator):

• 4th root of 67 = 67^(1/4)
• 67^(1/4) = 67^0.25
• = 2.861006

You can verify this works: 2.861006 × 2.861006 × 2.861006 × 2.861006 ≈ 67

This exponent relationship is why roots and powers are so closely connected—they're just two sides of the same coin.

With the formula in mind, using the calculator becomes even simpler.

Using the nth root calculator takes just a few seconds:

1. Enter the Number (Radicand): Type the number you want to find the root of in the first field. This can be any positive number—whole numbers, decimals, whatever you need.
2. Enter the Root Degree: In the second field, enter which root you want. Type 2 for square root, 3 for cube root, 4 for fourth root, and so on.
3. View Your Result: The calculator instantly displays your nth root value, calculated to six decimal places for precision.

That's it. No buttons to click—the result updates automatically as you type.

Some roots come out to clean whole numbers. These "perfect roots" are handy to know, especially for quick mental math checks:

When your result comes out to a clean whole number, you've found a perfect nth root. Memorizing a few of these can speed up your mental math significantly—you'll know right away when an answer should be exact versus a decimal.

Let's see how these calculations look in practice.

Here are some worked examples showing the calculator in action:

• Input: Number = 67, Root degree = 4
• Result: 2.861006
• Verification: 2.861006⁴ ≈ 67 ✓

This is the example shown in our calculator—a typical case where the answer isn't a clean whole number.

• Input: Number = 16, Root degree = 4
• Result: 2
• Verification: 2 × 2 × 2 × 2 = 16 ✓

A perfect 4th root, giving a clean whole number result.

• Input: Number = 27, Root degree = 3
• Result: 3
• Verification: 3 × 3 × 3 = 27 ✓

• Input: Number = 243, Root degree = 5
• Result: 3
• Verification: 3 × 3 × 3 × 3 × 3 = 243 ✓

• Input: Number = 50, Root degree = 2
• Result: 7.071068
• Verification: 7.071068² ≈ 50 ✓

Not all square roots are nice round numbers—and that's perfectly fine. The calculator handles both cases equally well.

Sometimes you might need to work through a root calculation by hand. Here's how to approach that.

If you ever need to estimate an nth root without a calculator, here's a practical approach:

Method: Use the Exponent Relationship

Since ⁿ√x = x^(1/n), you can work with exponents instead of roots:

1. Convert the root to a decimal exponent (1 ÷ n)
2. Use logarithms or estimation to calculate

Example: Find the 4th root of 80

1. 4th root = exponent of 1/4 = 0.25
2. 80^0.25 = ?
3. We know 2⁴ = 16 and 3⁴ = 81
4. Since 80 is just under 81, the 4th root of 80 is just under 3
5. Estimate: approximately 2.99 (actual: 2.9907)

For most practical purposes, bracketing between known perfect roots gives you a solid estimate. The calculator handles the precision—this method just helps you sanity-check results or work through problems when you don't have access to a tool.

Speaking of tools, if you need to find nth roots on a physical calculator, the next section has you covered.

Many scientific calculators have nth root functions, but finding them isn't always obvious. Here's how to access them on popular models:

1. Press MATH
2. Select option 5: ˣ√ (or scroll to find it)
3. Enter the root index first (like 4 for 4th root)
4. Press MATH and select 5 again
5. Enter your number
6. Press ENTER

1. Press SHIFT + ^ (or the x^y key) to access the root function
2. Enter the root degree
3. Enter your number
4. Press =

Use the exponent key since ⁿ√x = x^(1/n):

1. Enter your number
2. Press the exponent key (^ or y^x)
3. Enter 1 ÷ n as your exponent (use parentheses: 1/4 for 4th root)
4. Press equals

For example, 4th root of 67: Enter 67 ^ (1/4) = to get 2.861006

This exponent method is actually the most reliable since it works on any calculator that has a power function—no need to hunt for a specific nth root button.

You might wonder when higher roots actually come up outside of a classroom. They're more common than you'd expect:

Finance and Investing Compound Annual Growth Rate (CAGR) uses nth roots. If an investment grew from $10,000 to $15,000 over 5 years, the CAGR calculation involves a 5th root.

Engineering and Design When you know the volume of a cube but need the side length, that's a cube root. For more complex shapes or calculations, higher roots appear in formulas.

Statistics Geometric mean calculations—used when averaging rates or percentages—require nth roots where n equals the number of values being averaged.

Physics Various physics formulas involve roots beyond the square root, particularly in areas dealing with wave functions, decay rates, and dimensional analysis.

For everyday situations, square and cube roots handle most needs. But when you encounter that 4th, 5th, or 8th root, you'll be glad this calculator exists.

This calculator displays results to six decimal places, which is more than enough precision for virtually all practical applications. For most real-world calculations—homework, engineering estimates, financial projections—six decimal places exceeds what you'll need.

If you're doing highly specialized work requiring greater precision, the same formula (x^(1/n)) can be computed in scientific software or programming languages that support arbitrary precision.

Got a root to calculate? Enter your numbers above and you'll have your answer in seconds.