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nth Root

nth Root

The number that must be multiplied times itself n times to equal a given value. The nth root of x is written The nth root of x, written as a radical symbol with index n and radicand x or The nth root of x written as a fraction exponent: x to the power of 1/n. For example, The fifth root of 32 equals 2 since 25 = 32.

Notes: When n = 2 an nth root is called a square root. Also, if n is even and x is negative, then The nth root of x, written as a radical symbol with index n and radicand x is nonreal.

 

 

See also

Root rules, radical, radicand

Key Formula

xn=x 1/n\sqrt[n]{x} = x^{\,1/n}
Where:
  • xx = The radicand — the number you want to find the root of
  • nn = The index — how many times the root must be multiplied by itself to give x

Worked Example

Problem: Find the 4th root of 81.
Step 1: Write the problem using radical notation.
814=  ?\sqrt[4]{81} = \;?
Step 2: Ask: what number, raised to the 4th power, equals 81? Try 3.
34=3×3×3×3=813^4 = 3 \times 3 \times 3 \times 3 = 81
Step 3: Since 3 raised to the 4th power equals 81, the 4th root of 81 is 3. You can also express this with a fractional exponent.
814=811/4=3\sqrt[4]{81} = 81^{1/4} = 3
Answer: The 4th root of 81 is 3.

Another Example

Problem: Simplify the 5th root of 32.
Step 1: Write the expression.
325=  ?\sqrt[5]{32} = \;?
Step 2: Determine which number, multiplied by itself 5 times, gives 32. Try 2.
25=2×2×2×2×2=322^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32
Step 3: Confirm the result.
325=321/5=2\sqrt[5]{32} = 32^{1/5} = 2
Answer: The 5th root of 32 is 2.

Frequently Asked Questions

What is the difference between an nth root and an exponent?
An nth root is the inverse operation of raising to the nth power. If you know that an=xa^n = x, then taking the nth root recovers aa: xn=a\sqrt[n]{x} = a. Exponents build up a value by repeated multiplication; roots undo that process.
Can you take an even root of a negative number?
Not within the real numbers. For example, −164\sqrt[4]{-16} has no real answer because no real number raised to the 4th power gives a negative result (even powers always produce non-negative values). However, odd roots of negative numbers are real: −273=−3\sqrt[3]{-27} = -3 because (−3)3=−27(-3)^3 = -27.

Square root vs. nth root

A square root is simply an nth root where n=2n = 2. Writing x\sqrt{x} is shorthand for x2\sqrt[2]{x}. The nth root generalizes the idea to any positive integer index — cube roots (n=3n = 3), 4th roots (n=4n = 4), and so on. All the same rules about radicands and even/odd indices apply; the square root is just the most common special case.

Why It Matters

Nth roots appear throughout algebra and science whenever you need to reverse an exponent. For example, the compound interest formula requires an nth root to solve for an interest rate, and scientists use cube roots when computing volumes or densities. Mastering nth roots also prepares you for working with rational (fractional) exponents, which are essential in calculus and higher mathematics.

Common Mistakes

Mistake: Thinking that the nth root of a negative number is always undefined.
Correction: Only even roots of negative numbers are nonreal. Odd roots of negative numbers are perfectly valid real numbers. For example, −83=−2\sqrt[3]{-8} = -2.
Mistake: Confusing xn\sqrt[n]{x} with x/nx / n or x⋅nx \cdot n.
Correction: The nth root asks "what number raised to the nnth power gives xx?" It is equivalent to x1/nx^{1/n}, not division or multiplication by nn. For instance, 273=3\sqrt[3]{27} = 3, not 27/3=927/3 = 9.

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