Math Made Simple

nth Root Calculator: Generalizing Roots for Any Power

Learn one flexible method for square roots, cube roots, fourth roots, and beyond.

The big idea

An nth root asks which number was multiplied by itself n times. If N = aⁿ, then the nth root of N is a. Square roots use n = 2, cube roots use n = 3, and fourth roots use n = 4.

The general formula

If N = aⁿ, then ⁿ√N = a

On a small phone screen, swipe horizontally inside the formula box if the equation feels wide.

The four-step nth-root trick

1. Identify the index. The small number on the root symbol tells you the power. For a square root, the index is 2 and is usually left unwritten.
2. Estimate the answer. Find nearby powers such as 4³ = 64 and 5³ = 125.
3. Match the exact power. Look for the number that becomes the radicand after being multiplied by itself n times.
4. Verify. Raise your proposed answer to the index power. The result must equal the original number.

A useful pattern

Roots and powers undo each other when the number is in the appropriate domain. For a nonnegative base a:

ⁿ√(aⁿ) = a

For even indices, this principal-root rule uses a nonnegative a. For odd indices, negative bases are allowed; for example, ∛(−8) = −2.

Worked examples

Example 1: A square root

The index is 2. Since 12 × 12 = 144, reverse the power:

√144 = 12 because 12² = 144

Example 2: A cube root

The index is 3. Because 7 × 7 × 7 = 343:

∛343 = 7 because 7³ = 343

Example 3: A fourth root

The index is 4. Since 3² = 9 and 9² = 81, we have 3⁴ = 81.

⁴√81 = 3 because 3⁴ = 81

Example 4: A fifth root

The index is 5. Calculate 2 × 2 × 2 × 2 × 2 = 32, so:

⁵√32 = 2 because 2⁵ = 32

Common powers to remember

Base	Square	Cube	Fourth power
2	4	8	16
3	9	27	81
4	16	64	256
5	25	125	625
10	100	1,000	10,000

On a small phone screen, swipe horizontally inside the table to view every column.

Memory tip

“Read the index, reverse the power, then verify.” The index tells you how many equal factors to use. For example, the 4 in ⁴√625 means find the number whose fourth power is 625. Since 5⁴ = 625, the answer is 5.

Common mistakes to avoid

• Do not ignore the index. √64 and ∛64 are different: √64 = 8, while ∛64 = 4.
• Do not multiply the index by the radicand. An nth root asks for repeated multiplication, not n × N.
• Do not assume every number has a whole-number nth root. For example, 20 is not a perfect cube.
• For an even index, a negative radicand has no real root. For example, there is no real value for ⁴√(−16).
• For an odd index, keep the sign. ∛(−27) = −3 because (−3)³ = −27.

Practice questions

Find each root and verify it by raising your answer to the indicated power.

1. √225 = ?
2. ∛512 = ?
3. ⁴√256 = ?
4. ⁵√3,125 = ?
5. Is 100 a perfect cube?

Answers

1. √225 = 15, because 15² = 225.
2. ∛512 = 8, because 8³ = 512.
3. ⁴√256 = 4, because 4⁴ = 256.
4. ⁵√3,125 = 5, because 5⁵ = 3,125.
5. 100 is not a perfect cube. It lies between 4³ = 64 and 5³ = 125.

Final takeaway

The nth-root method works for any positive integer index: read the index, find the number that produces the radicand after repeated multiplication, and verify by raising the answer to that power. This one idea connects square roots, cube roots, fourth roots, and all higher roots.