Nth Roots

Extend your understanding of roots beyond square and cube roots to any index.

Advanced25 minLesson
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Definition

An nth root of a number aa is a value that, when raised to the nnth power, gives aa.
We write the nth root as:
an=bif and only ifbn=a\sqrt[n]{a} = b \quad \text{if and only if} \quad b^n = a
The number nn is called the index of the radical.
Special cases:
  • a2=a\sqrt[2]{a} = \sqrt{a} (square root, index 2 is usually omitted)
  • a3\sqrt[3]{a} (cube root)
  • a4\sqrt[4]{a} (fourth root)
  • a5\sqrt[5]{a} (fifth root)
Key property:
an=a1/n\sqrt[n]{a} = a^{1/n}
This means nth roots can be written as fractional exponents!

Try it now

What is 273\sqrt[3]{27}?

Worked Examples

Evaluate 814\sqrt[4]{81}
1

Understand what we need

Find a number that, when raised to the 4th power, equals 81 → b4=81b^4 = 81
2

Think about powers of small numbers

24=162^4 = 16, 34=813^4 = 81 \checkmark → 34=813^4 = 81
3

Verify

3×3×3×3=9×9=813 \times 3 \times 3 \times 3 = 9 \times 9 = 81 \checkmark → 814=3\sqrt[4]{81} = 3

Common Mistakes

Thinking −164=−2\sqrt[4]{-16} = -2 (even root of a negative)

Why it's wrong: Even roots of negative numbers are not real. (−2)4=16(-2)^4 = 16, not −16-16. You need four factors, and an even number of negatives gives a positive result.

Correct: −164\sqrt[4]{-16} is not a real number. Only odd roots of negatives exist in real numbers.

Confusing the index with the exponent inside: x33=x9\sqrt[3]{x^3} = x^9

Why it's wrong: The index goes in the denominator, not multiplied: xmn=xm/n\sqrt[n]{x^m} = x^{m/n}, not xmnx^{mn}.

Correct: x33=x3/3=x1=x\sqrt[3]{x^3} = x^{3/3} = x^1 = x

Forgetting that ann=∣a∣\sqrt[n]{a^n} = |a| for even nn

Why it's wrong: For even indices, we take the principal (positive) root. (−2)44=164=2\sqrt[4]{(-2)^4} = \sqrt[4]{16} = 2, not −2-2.

Correct: ann=a\sqrt[n]{a^n} = a if nn is odd, but ann=∣a∣\sqrt[n]{a^n} = |a| if nn is even.

Interactive Visual

Simplifying Radicals

484\sqrt[4]{48}
Simplify the root: look for the largest factor under the root that you can take the root of.

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is 273\sqrt[3]{27}?

Why It Matters

Nth roots appear throughout advanced mathematics and real-world applications:
  • Finance: Compound annual growth rate (CAGR) uses nth roots: FinalInitialn−1\sqrt[n]{\frac{\text{Final}}{\text{Initial}}} - 1
  • Statistics: Geometric mean of nn values uses the nth root
  • Physics: Scaling laws often involve fourth and higher roots
  • Engineering: Root-mean-square calculations in electrical engineering
  • Geometry: Volume scaling between similar solids
Understanding nth roots gives you the tools to work with any radical expression!

Real World Applications

Investment Growth Rate

Financial analysts use nth roots to calculate compound annual growth rates for investments held over multiple years.

Example:

If an investment doubles in 7 years, the annual rate is 27−1≈10.4%\sqrt[7]{2} - 1 \approx 10.4\%

1Try It Yourself

An investment of 5000 dollars grows to 8000 dollars in 4 years.

What is the annual growth rate?

Step 1: Write the mathematical expression

Calculate 8000/50004−1\sqrt[4]{8000/5000} - 1:

Geometric Mean in Statistics

The geometric mean of $n$ values is the nth root of their product, used for averaging ratios and percentages.

Example:

The geometric mean of 2, 4, 8 is 2×4×83=643=4\sqrt[3]{2 \times 4 \times 8} = \sqrt[3]{64} = 4

2Try It Yourself

A company's revenue grew by factors of 1.2, 1.5, and 1.8 over three consecutive years.

What is the average growth factor?

Step 1: Write the mathematical expression

Find 1.2×1.5×1.83\sqrt[3]{1.2 \times 1.5 \times 1.8}:

Key Takeaways

  • 1an=b\sqrt[n]{a} = b means bn=ab^n = a (b is the nth root of a)
  • 2Nth roots can be written as fractional exponents: an=a1/n\sqrt[n]{a} = a^{1/n}
  • 3The power rule: amn=am/n\sqrt[n]{a^m} = a^{m/n}
  • 4Odd roots of negative numbers are real; even roots of negatives are not real
  • 5a×bn=an×bn\sqrt[n]{a \times b} = \sqrt[n]{a} \times \sqrt[n]{b} (product rule for radicals)

Frequently Asked Questions

No, not in real numbers. Even roots (2nd, 4th, 6th, etc.) of negative numbers don't exist as real numbers because any real number raised to an even power is positive.
No, not in real numbers. Even roots (2nd, 4th, 6th, etc.) of negative numbers don't exist as real numbers because any real number raised to an even power is positive.
They are exactly the same! Both equal 2. The radical notation an\sqrt[n]{a} and the fractional exponent a1/na^{1/n} are two ways of writing the same thing.
64=2664 = 2^6, so 646=266=2\sqrt[6]{64} = \sqrt[6]{2^6} = 2. Alternatively, 641/6=(26)1/6=21=264^{1/6} = (2^6)^{1/6} = 2^1 = 2.

Glossary

Nth root
A value that, when raised to the nth power, gives the original number
Index
The small number in the radical symbol that indicates which root to take (n in an\sqrt[n]{a})
Radicand
The number under the radical sign (a in an\sqrt[n]{a})
Principal root
The non-negative real root when the index is even
Fractional exponent
An exponent written as a fraction, where a1/n=ana^{1/n} = \sqrt[n]{a}
Today's challengeOne problem, three tries

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