nth Root
In short: An nth root undoes the nth power: it is the number which, used as a factor n times, rebuilds the original, and it is written under a radical with index n.
Algebra 2Numbers and OperationsN-RN.A.2
What is nth Root?
The nth root of a number \(a\), written \(\sqrt[n]{a}\), undoes raising to the \(n\)th power: \(\sqrt[n]{a} = b\) means \(b^n = a\). The small number \(n\) is the index, so \(n = 2\) is the familiar square root and \(n = 3\) is the cube root.
Parity matters. An odd index accepts any real radicand, so \(\sqrt[3]{-216} = -6\). An even index needs a non-negative radicand to stay real, and the symbol returns the non-negative answer. In exponent form, \(\sqrt[n]{a} = a^{1/n}\).
How to spot it
- A story asks for the side of a cube given its capacity, or a yearly growth rate given a multi-year total.
- A small number sits in the crook of a radical sign, as in \(\sqrt[5]{32}\), signaling an nth root.
- An equation such as \(x^5 = 243\) or \(x^4 = 81\) must be solved for \(x\).
- A fractional exponent such as \(\frac{1}{3}\) or \(\frac{1}{4}\) appears on a base.
The method
- To find an nth root, read the index \(n\) and the radicand \(a\), then ask which factor repeated \(n\) times gives \(a\).
- When it is not obvious, factor the radicand into perfect \(n\)th powers, such as \(16x^8 = 2^4(x^2)^4\).
- Check the sign: an odd index keeps the sign of the radicand; an even index gives non-negative answers, and equations may need both signs.
- Verify by raising the result to the \(n\)th power to land back on the radicand.
nth Root: solved example
Problem
A cube-shaped toolbox holds exactly 343 cubic inches. How long is each edge? Afterward, solve \(x^4 = 81\).
Solution
The edge satisfies \(e^3 = 343\), so \(e = \sqrt[3]{343}\). Since \(7 \times 7 \times 7 = 343\), the edge is 7 inches.
For \(x^4 = 81\), the index is even, so both signs work: \(3^4 = 81\) and \((-3)^4 = 81\). The edge is 7 inches and the solutions are \(x = \pm 3\).
Try it
Question 1
Evaluate \(\sqrt[3]{-125}\).
Show the answer
The index is odd, so a negative radicand is fine: \((-5)^3 = -125\), giving \(-5\).
Question 2
Evaluate \(\sqrt[6]{64}\).
Show the answer
\(2^6 = 64\), so the sixth root is 2.
nth Root vs. Square Root
The square root is the case where the index is 2, and it is usually written without the index. The general version includes the cube root, fourth root and beyond. A negative radicand has a real cube root but no real square root, so the odd or even index decides what is allowed.
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