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Root of a Number — Definition, Formula & Examples

Root of a Number

A term that can refer to the square root or nth root of a number.

 

 

See also

Radical rules

Key Formula

an=bmeansbn=a\sqrt[n]{a} = b \quad \text{means} \quad b^n = a
Where:
  • aa = The number you are taking the root of (the radicand)
  • nn = The index of the root — it tells you which root to find (2 for square root, 3 for cube root, etc.)
  • bb = The root itself — the value that, when raised to the nth power, equals a

Worked Example

Problem: Find the cube root of 64.
Step 1: Write the problem using radical notation.
643=  ?\sqrt[3]{64} = \;?
Step 2: Ask: what number, raised to the 3rd power, equals 64?
b3=64b^3 = 64
Step 3: Test b = 4.
43=4×4×4=64  ✓4^3 = 4 \times 4 \times 4 = 64 \; \checkmark
Answer: The cube root of 64 is 4, so √[3]{64} = 4.

Another Example

Problem: Find the 4th root of 81.
Step 1: Write the problem in radical form.
814=  ?\sqrt[4]{81} = \;?
Step 2: Ask: what number, raised to the 4th power, equals 81?
b4=81b^4 = 81
Step 3: Test b = 3.
34=3×3×3×3=81  ✓3^4 = 3 \times 3 \times 3 \times 3 = 81 \; \checkmark
Answer: The 4th root of 81 is 3, so √[4]{81} = 3.

Frequently Asked Questions

What is the difference between a root and a power?
A power (exponent) and a root are inverse operations. Raising 3 to the power of 2 gives 9; taking the square root of 9 gives back 3. In general, if bn=ab^n = a, then an=b\sqrt[n]{a} = b. Think of roots as "undoing" exponents.
Can you take the root of a negative number?
It depends on the index. Odd roots of negative numbers are defined and negative: for example, −83=−2\sqrt[3]{-8} = -2 because (−2)3=−8(-2)^3 = -8. Even roots (square root, 4th root, etc.) of negative numbers are not real numbers because no real number raised to an even power gives a negative result.

Square Root vs. nth Root

A square root is just the specific case where the index n=2n = 2. The nth root is the general concept: you can choose any positive integer for nn. When someone writes a\sqrt{a} without an index, it always means the square root (n=2n = 2). Writing an\sqrt[n]{a} with an explicit index covers cube roots (n=3n = 3), fourth roots (n=4n = 4), and beyond.

Why It Matters

Roots appear throughout algebra, geometry, and science. You use square roots when solving quadratic equations and applying the Pythagorean theorem. Cube roots arise when working with volumes — for instance, finding the side length of a cube given its volume. Understanding roots also prepares you for fractional exponents, since an=a1/n\sqrt[n]{a} = a^{1/n}, a key idea in higher math.

Common Mistakes

Mistake: Confusing 273\sqrt[3]{27} with 27÷327 \div 3, thinking "cube root" means dividing by 3.
Correction: A cube root asks what number multiplied by itself three times equals 27. That number is 3, because 3×3×3=273 \times 3 \times 3 = 27. Division by the index is not what roots do.
Mistake: Assuming the square root of a negative number is just a negative number, e.g., writing −25=−5\sqrt{-25} = -5.
Correction: Check: (−5)2=25(-5)^2 = 25, not −25-25. The square root of a negative number is not a real number. Even roots of negative numbers require imaginary numbers (like 5i5i), which are a separate topic.

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