nth Roots: Evaluate, Simplify and Solve

nth roots undo raising to the nth power. The nth root of $a$ , written $\sqrt[n]{a}$ , is a number whose nth power is $a$ . For example, $\sqrt[3]{8} = 2$ because $2^3 = 8$ . Odd roots exist for every real number, but even roots of negative numbers are not real.

Before this lesson: You should know Square Root, Cube Roots, Simplifying Square Roots and Exponent Rules.

What an nth Root Is

A square root undoes squaring, and a cube root undoes cubing. An nth root undoes raising to any power $n$ . If $b^n = a$ , then $b$ is an nth root of $a$ .

In $\sqrt[n]{a}$ , the small number $n$ is the index and $a$ is the radicand. A square root has index 2, which we usually don’t write.

Rules for Real nth Roots

Index	Radicand	Real nth roots	Example
Odd	Any real number	Exactly one	$\sqrt[3]{-27} = -3$
Even	Positive	Two, $\pm$ ; the symbol means the positive one	$\sqrt[4]{16} = 2$ , and −2 is also a 4th root of 16
Even	Negative	None	$\sqrt[4]{-16}$ is not a real number

The positive root is called the principal root. The radical symbol always means the principal root.

Two graphs side by side. Left: y = x^3 and the horizontal line y = 8 meet at one point, x = 2, so x^3 = 8 has one real solution. Right: y = x^4 and the line y = 16 meet at two points, x = −2 and x = 2, so x^4 = 16 has two real solutions. — An odd power hits each height once. An even power hits each positive height twice, at ± the root.

Two rules help you simplify, when the roots are real:

\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b} \qquad \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}

Also, $\sqrt[n]{a^n} = a$ when $n$ is odd, and $\sqrt[n]{a^n} = |a|$ when $n$ is even.

Worked Examples

Example 1: Evaluate nth roots

Evaluate (a) $\sqrt[3]{-64}$ , (b) $\sqrt[4]{81}$ , (c) $\sqrt[5]{32}$ and (d) $\sqrt[4]{-16}$ .

(a) Ask what cubed is −64: $(-4)^3 = -64$ , so $\sqrt[3]{-64} = -4$
(b) Ask what positive number to the 4th is 81: $3^4 = 81$ , so $\sqrt[4]{81} = 3$
(c) Ask what to the 5th is 32: $2^5 = 32$ , so $\sqrt[5]{32} = 2$
(d) Any real number to the 4th power is 0 or positive, so it can’t be −16.

Answer: (a) −4 (b) 3 (c) 2 (d) not a real number

Example 2: Simplify by pulling out perfect powers

Simplify (a) $\sqrt[3]{54}$ and (b) $\sqrt[4]{48}$ .

(a) Find the largest perfect cube factor of 54 (see prime factorization): $54 = 27 \cdot 2$
Split the root (see simplifying square roots for the same idea): $\sqrt[3]{27} \cdot \sqrt[3]{2}$
Take the cube root of 27: $3\sqrt[3]{2}$
(b) Find the largest perfect 4th power factor of 48: $48 = 16 \cdot 3$
Split the root: $\sqrt[4]{16} \cdot \sqrt[4]{3}$
Take the 4th root of 16: $2\sqrt[4]{3}$

Answer: (a) $3\sqrt[3]{2}$ (b) $2\sqrt[4]{3}$

Example 3: Variables under an nth root

Simplify $\sqrt[3]{16x^5y^6}$ .

Split each factor into a perfect cube and a leftover: $16 = 8 \cdot 2$ , $x^5 = x^3 \cdot x^2$ , $y^6 = (y^2)^3$
Group the perfect cubes: $\sqrt[3]{8x^3y^6} \cdot \sqrt[3]{2x^2}$
Take the cube root of each perfect cube (see exponent rules): $2xy^2 \cdot \sqrt[3]{2x^2}$

Answer: $2xy^2\sqrt[3]{2x^2}$

With an even index, watch the sign. $\sqrt[4]{81x^4} = 3|x|$ , because $x$ could be negative but the 4th root can’t be (see absolute value).

Example 4: Solve equations with powers

Solve (a) $x^3 = -125$ , (b) $x^4 = 625$ and (c) $2x^5 + 3 = 67$ .

(a) Take the cube root of both sides: $x = \sqrt[3]{-125}$
Evaluate: $x = -5$
(b) Take the 4th root of both sides. An even power needs $\pm$ (see solving quadratics with square roots): $x = \pm\sqrt[4]{625}$
Evaluate: $x = \pm 5$
(c) Subtract 3 from both sides: $2x^5 = 64$
Divide both sides by 2: $x^5 = 32$
Take the 5th root: $x = 2$

Answer: (a) $x = -5$ (b) $x = 5$ or $x = -5$ (c) $x = 2$

Example 5: Word problem with average growth

An investment grew from $1,000 to $1,331 in 3 years. It grew by the same percent each year. Find that yearly growth rate.

Let the yearly growth factor be $g$ . After 3 years: $1000g^3 = 1331$
Divide both sides by 1000: $g^3 = 1.331$
Take the cube root: $g = \sqrt[3]{1.331}$
Evaluate, since $1.1^3 = 1.331$ : $g = 1.1$
The growth factor 1.1 means $1.1 - 1 = 0.1$ growth each year (see percentages).

Answer: The investment grew 10% per year. Check: $1000 \cdot 1.1 = 1100$ , $1100 \cdot 1.1 = 1210$ , $1210 \cdot 1.1 = 1331$ .

Common Mistakes

Mistake	Example of the error	Fix
Taking an even root of a negative	$\sqrt[4]{-16} = -2$	$(-2)^4 = 16$ , not −16. The 4th root of −16 is not real.
Thinking odd roots of negatives don’t exist	$\sqrt[3]{-27}$ is undefined	$(-3)^3 = -27$ , so $\sqrt[3]{-27} = -3$ .
Forgetting ± when solving	$x^4 = 16$ , so $x = 2$	Even powers give two real solutions: $x = \pm 2$ .
Dividing the radicand by the index	$\sqrt[3]{27} = 9$	The cube root asks what cubed is 27: $3^3 = 27$ , so the answer is 3.
Pulling out a non-perfect power	$\sqrt[3]{54} = 2\sqrt[3]{27}$	Pull out the perfect cube: $\sqrt[3]{27 \cdot 2} = 3\sqrt[3]{2}$ .

Practice Problems

Evaluate $\sqrt[4]{625}$ .
Evaluate $\sqrt[3]{-216}$ .
Evaluate $\sqrt[5]{-243}$ .
Is $\sqrt[6]{-64}$ a real number? Explain.
Simplify $\sqrt[3]{40}$ .
Simplify $\sqrt[4]{162}$ .
Simplify $\sqrt[3]{-54a^4}$ . Assume $a$ is positive.
Simplify $\sqrt[4]{16x^8y^5}$ . Assume all variables are positive.
Solve $x^4 - 7 = 74$ .
A town’s population grew from 2,000 to 3,456 in 3 years, by the same percent each year. Find the yearly growth rate.

Show answers

Ask what to the 4th is 625: $5^4 = 625$
Answer: 5
Ask what cubed is −216: $(-6)^3 = -216$
Answer: −6
Ask what to the 5th is −243: $(-3)^5 = -243$
Answer: −3
The index 6 is even, and no real number to an even power is negative
Answer: no, it is not real
Perfect cube factor: $40 = 8 \cdot 5$
Split: $\sqrt[3]{8} \cdot \sqrt[3]{5}$
Answer: $2\sqrt[3]{5}$
Perfect 4th power factor: $162 = 81 \cdot 2$
Split: $\sqrt[4]{81} \cdot \sqrt[4]{2}$
Answer: $3\sqrt[4]{2}$
Perfect cubes: $-54a^4 = (-27a^3)(2a)$
Split: $\sqrt[3]{-27a^3} \cdot \sqrt[3]{2a}$
Answer: $-3a\sqrt[3]{2a}$
Perfect 4th powers: $16x^8y^5 = (16x^8y^4)(y)$
Split: $\sqrt[4]{16x^8y^4} \cdot \sqrt[4]{y}$
Answer: $2x^2y\sqrt[4]{y}$
Add 7: $x^4 = 81$
Take the 4th root with $\pm$ : $x = \pm\sqrt[4]{81}$
Answer: $x = \pm 3$
Growth factor equation: $2000g^3 = 3456$
Divide by 2000: $g^3 = 1.728$
Cube root, since $1.2^3 = 1.728$ : $g = 1.2$
Subtract 1: $1.2 - 1 = 0.2$
Answer: 20% per year

FAQ

What is an nth root?

The nth root of $a$ is a number that gives $a$ when raised to the nth power. For example, $\sqrt[4]{81} = 3$ because $3^4 = 81$ .

Can you take the nth root of a negative number?

Yes, if $n$ is odd: $\sqrt[5]{-32} = -2$ . If $n$ is even, the root of a negative number is not a real number.

How do you simplify an nth root?

Write the radicand as a product with the largest perfect nth power you can find. Take the root of that part, and leave the rest under the radical.

What is the principal nth root?

When $n$ is even and $a$ is positive, $a$ has two real nth roots, one positive and one negative. The principal root is the positive one, and it is what $\sqrt[n]{a}$ means.

Summary

$\sqrt[n]{a} = b$ means $b^n = a$ ; $n$ is the index.
Odd roots exist for every real number and keep its sign.
Even roots of negative numbers are not real; even roots of positive numbers mean the positive root.
Simplify by pulling out perfect nth powers.
When solving $x^n = k$ with $n$ even and $k$ positive, include $\pm$ .

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