Cube Roots

Q: Is a cube root always smaller than the number inside?

For numbers bigger than $1$, yes — $\sqrt[3]{1000} = 10$. But between $0$ and $1$ it flips: $\sqrt[3]{\dfrac{1}{8}} = \dfrac{1}{2}$, which is larger than $\dfrac{1}{8}$. Don't assume; check with the definition.

Cubing a number means using it as a factor three times: $4^3 = 4 \cdot 4 \cdot 4 = 64$ . A cube root runs that in reverse — $\sqrt[3]{64}$ asks, what number used as a factor three times makes $64$ ? The answer is $4$ .

Cube roots show up any time volume is involved, because a cube's volume is its edge used as a factor three times. Know your perfect cubes and the whole topic becomes quick recall plus one big idea: unlike square roots, cube roots are perfectly happy with negative numbers.

Perfect cubes to know

A perfect cube is what you get when you cube a whole number. The first ten are worth memorizing: $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000$ — from $1^3$ up through $10^3$ .

If the number under the root is on that list, the cube root is a whole number: $\sqrt[3]{27} = 3$ , $\sqrt[3]{125} = 5$ , $\sqrt[3]{1000} = 10$ . If it isn't, the cube root is irrational and you estimate it instead.

Reading the notation

The small $3$ tucked into the root symbol is the index: $\sqrt[3]{64}$ means cube root, while a plain $\sqrt{64}$ means square root. The two give different answers — $\sqrt[3]{64} = 4$ because $4 \cdot 4 \cdot 4 = 64$ , but $\sqrt{64} = 8$ because $8 \cdot 8 = 64$ .

Taking a cube root is the opposite of cubing, the same way taking a square root is the opposite of squaring. To evaluate one, hunt for the number that gives the target when used as a factor three times.

Negatives are allowed

A negative number times itself three times stays negative: $(-3)(-3)(-3) = -27$ . Two of the negatives cancel, and the third keeps the sign. So every negative number has a real cube root, and that root is negative: $\sqrt[3]{-27} = -3$ .

Compare that with square roots, where $\sqrt{-25}$ has no real answer because no real number times itself gives a negative. This is the sharpest difference between the two roots, and test questions love it.

Worked examples

Example 1: a perfect cube

Evaluate $\sqrt[3]{64}$ .

Ask: what number used as a factor three times makes

64

Test

4

4 \cdot 4 \cdot 4 = 64

Result

\sqrt[3]{64} = 4

Answer: $\sqrt[3]{64} = 4$

Example 2: a negative number

Evaluate $\sqrt[3]{-27}$ .

A negative number has a negative cube root

Test

-3

(-3)(-3)(-3) = -27

Result

\sqrt[3]{-27} = -3

Answer: $\sqrt[3]{-27} = -3$

Example 3: working backward from volume

A cube-shaped box has a volume of $216$ cubic inches. How long is each edge?

The edge used as a factor three times gives the volume

e^3 = 216

Take the cube root of both sides

e = \sqrt[3]{216}

Test

6

6 \cdot 6 \cdot 6 = 216

Each edge is

6

inches

Answer: $e = 6$ inches

Try one yourself

Common questions

What's the difference between a square root and a cube root?

A square root reverses using a number as a factor twice; a cube root reverses using it three times. $\sqrt{64} = 8$ because $8 \cdot 8 = 64$ , while $\sqrt[3]{64} = 4$ because $4 \cdot 4 \cdot 4 = 64$ . Always check the small index number in the root symbol.

Why can I take the cube root of a negative but not the square root?

Sign rules. A number times itself an even number of times can never be negative, so $\sqrt{-25}$ has no real answer. But three negative factors leave one negative sign standing: $(-4)(-4)(-4) = -64$ , so $\sqrt[3]{-64} = -4$ .

Is a cube root always smaller than the number inside?

For numbers bigger than $1$ , yes — $\sqrt[3]{1000} = 10$ . But between $0$ and $1$ it flips: $\sqrt[3]{\dfrac{1}{8}} = \dfrac{1}{2}$ , which is larger than $\dfrac{1}{8}$ . Don't assume; check with the definition.

What if the number isn't a perfect cube?

Then the cube root is irrational, and you trap it between the perfect cubes on either side. For $\sqrt[3]{100}$ : since $64 < 100 < 125$ , the root is between $4$ and $5$ .

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Cube Roots

Perfect cubes to know

Reading the notation

Negatives are allowed

Worked examples

Example 1: a perfect cube

Example 2: a negative number

Example 3: working backward from volume

Try one yourself

Common questions

Keep learning

Perfect Squares & Square Roots

Estimating Irrational Roots

Classifying Real Numbers