Key takeaway: The nth root of a number is the value that, when raised to the power n, returns the original number, written with a radical sign and an index n, or as a rational exponent 1/n.
Grade 11 – Algebra 2N-RN.A.2Numbers and Operations
Definition of nth Root
The nth root of a number \(a\) reverses raising to the power \(n\). Since \(6^4 = 1296\), the fourth root of 1296 is 6. Square roots have index 2 and cube roots index 3, and the index tells how many equal factors multiply to give the number.
The parity of the index matters. An odd root exists for every real number: \(\sqrt[3]{-216} = -6\). An even root of a negative number is not real, and an even root of a positive number is taken as its positive value, called the principal root. The same idea can be written with a fractional exponent.
Formula and notation
Writing an nth root with index \(n\) and radicand \(a\):
\[ \sqrt[n]{a} = a^{1/n}, \qquad (\sqrt[n]{a})^n = a \]
Products split: \(\sqrt[n]{ab} = \sqrt[n]{a}\cdot\sqrt[n]{b}\) for real roots. A general power gives \(\sqrt[n]{a^m} = a^{m/n}\).
Key facts
- An odd nth root exists for negative radicands; an even one does not, in the reals.
- The principal even root is positive, so the fourth root of 81 is 3, not -3.
- The radical form and the exponent 1/n mean exactly the same thing.
- To simplify, pull out perfect nth powers from under the radical.
nth Root example
Evaluate \(\sqrt[3]{-216}\) and \(\sqrt[4]{1296}\), then simplify \(\sqrt[4]{48}\) and write \(32^{3/5}\) as an integer.
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\((-6)^3 = -216\), so \(\sqrt[3]{-216} = -6\). And \(6^4 = 1296\), so \(\sqrt[4]{1296} = 6\).
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Factor out a fourth power: \(\sqrt[4]{48} = \sqrt[4]{16\cdot 3} = 2\sqrt[4]{3}\).
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Fractional exponent: \(32^{3/5} = (\sqrt[5]{32})^3 = 2^3 = 8\).
The values are \(-6\), 6, \(2\sqrt[4]{3}\) and 8.
Common mistake
Watch out
Pat claims \(\sqrt[4]{-16} = -2\), because \((-2)^4\) looks like it should give back the negative number.
Why it is wrong
An even power of any real number is nonnegative, so \((-2)^4 = +16\), not \(-16\). No real number has a fourth power of \(-16\), so the real fourth root does not exist. It requires complex numbers.
Related terms and lessons
Browse the glossary: A–Z list · Algebra 2 vocabulary · Numbers and Operations