Cubes and Cube Roots

A cube of a number is that number multiplied by itself three times. A cube root reverses the process — it asks, “What number, when cubed, gives this value?” Cubes and cube roots come up whenever you work with three-dimensional measurements like volume, and they have one key difference from square roots: cube roots of negative numbers exist.

What Is a Perfect Cube?

A perfect cube is the result of multiplying a whole number by itself three times.

$n^3 = n \times n \times n$

Here are the perfect cubes you should know:

Example 1: Evaluate $4^3$

$4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64$

Answer: $4^3 = 64$

Example 2: Evaluate $6^3$

$6^3 = 6 \times 6 \times 6 = 36 \times 6 = 216$

Answer: $6^3 = 216$

Cube Root Notation

The cube root of a number $a$ is written with a radical symbol and a small 3 (called the index):

$\sqrt[3]{a} = b \quad \text{means} \quad b^3 = a$

We read $\sqrt[3]{a}$ as “the cube root of $a$.”

Example 3: Evaluate $\sqrt[3]{27}$

We need a number that, when cubed, gives 27.

$3 \times 3 \times 3 = 27 \quad \Rightarrow \quad \sqrt[3]{27} = 3$

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

Example 4: Evaluate $\sqrt[3]{125}$

$5 \times 5 \times 5 = 125 \quad \Rightarrow \quad \sqrt[3]{125} = 5$

Answer: $\sqrt[3]{125} = 5$

Example 5: Evaluate $\sqrt[3]{1{,}000}$

$10 \times 10 \times 10 = 1{,}000 \quad \Rightarrow \quad \sqrt[3]{1{,}000} = 10$

Answer: $\sqrt[3]{1{,}000} = 10$

Negative Cube Roots — They Exist!

Here is the big difference between square roots and cube roots. You cannot take the square root of a negative number (in real numbers), but you can take the cube root of a negative number.

Why? Because a negative number times a negative number times a negative number is negative:

$(-3) \times (-3) \times (-3) = 9 \times (-3) = -27$

Therefore:

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

This works for any negative perfect cube:

Example 6: Evaluate $\sqrt[3]{-64}$

$(-4)^3 = (-4) \times (-4) \times (-4) = 16 \times (-4) = -64$

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

Example 7: Evaluate $\sqrt[3]{-729}$

$(-9)^3 = (-9) \times (-9) \times (-9) = 81 \times (-9) = -729$

Answer: $\sqrt[3]{-729} = -9$

Why Square Roots of Negatives Fail but Cube Roots Succeed

It comes down to how many times you multiply:

• Squaring (two factors): $(-a) \times (-a) = +a^2$. Two negatives always make a positive, so you can never square a real number and get a negative result. That is why $\sqrt{-25}$ has no real answer.

• Cubing (three factors): $(-a) \times (-a) \times (-a) = +a^2 \times (-a) = -a^3$. The third factor flips the sign back to negative. So every negative number has a real cube root.

The Volume Connection

Cubes and cube roots have a natural connection to volume. A cube with side length $s$ has volume:

$V = s^3$

If you know the volume and want the side length:

$s = \sqrt[3]{V}$

Example 8: Side Length from Volume

A shipping box has a volume of 343 cubic inches and is a perfect cube. What is the side length?

$s = \sqrt[3]{343} = 7 \text{ inches}$

Verification: $7^3 = 7 \times 7 \times 7 = 343$. Checks out.

Answer: Each side is 7 inches.

Estimating Non-Perfect Cube Roots

Just like with square roots, you can estimate cube roots of numbers that are not perfect cubes.

Example 9: Estimate $\sqrt[3]{50}$

Step 1: Find the surrounding perfect cubes.

$3^3 = 27 \quad \text{and} \quad 4^3 = 64$

So $\sqrt[3]{50}$ is between 3 and 4.

Step 2: Determine where 50 falls between 27 and 64.

$\frac{50 - 27}{64 - 27} = \frac{23}{37} \approx 0.62$

Step 3: Estimate.

$\sqrt[3]{50} \approx 3 + 0.62 = 3.62$

Calculator check: $\sqrt[3]{50} \approx 3.684$. Our estimate is in the right ballpark (the linear interpolation is less accurate for cube roots, but it gives a reasonable starting point).

Real-World Application: Carpentry — Cubic Storage

A carpenter is building a cube-shaped storage bin that needs to hold 512 cubic feet of firewood. What should each side measure?

$s = \sqrt[3]{512} = 8 \text{ feet}$

Verification: $8 \times 8 \times 8 = 512$. Checks out.

Answer: Each side should be 8 feet.

The carpenter also knows the total surface area will be:

$\text{Surface Area} = 6 \times s^2 = 6 \times 64 = 384 \text{ square feet}$

This helps estimate the lumber needed.

Real-World Application: HVAC — Duct Sizing

An HVAC technician needs to replace a circular duct with a square duct of equal cross-sectional area. If the target airflow requires a duct with a cross-section of 64 square inches, the side of the square duct is:

$s = \sqrt{64} = 8 \text{ inches}$

But if the technician needs a cube-shaped plenum (junction box) with a volume of 216 cubic inches, the side length is:

$s = \sqrt[3]{216} = 6 \text{ inches}$

Understanding the difference between squares (for area) and cubes (for volume) is critical in the HVAC trade.

Comparing Squares vs. Cubes at a Glance

Common Mistakes to Avoid

1. Assuming cube roots of negative numbers do not exist. They do. $\sqrt[3]{-8} = -2$.

2. Confusing cubes with multiplication by 3. $4^3 = 64$, not $12$. Cubing means multiplying the number by itself three times, not multiplying by 3.

3. Mixing up square root and cube root notation. $\sqrt{64} = 8$ (square root), but $\sqrt[3]{64} = 4$ (cube root). The little index number matters.

4. Forgetting that cube growth is fast. $10^3 = 1{,}000$ while $10^2 = 100$. Volumes grow much faster than areas.

Practice Problems

Key Takeaways

• A perfect cube is $n^3 = n \times n \times n$. Memorize the cubes of 1 through 10.
• The cube root $\sqrt[3]{a}$ asks “what number cubed gives $a$?”
• Cube roots of negative numbers are real — unlike square roots. $\sqrt[3]{-8} = -2$.
• Cubes connect directly to volume: $V = s^3$ for a cube, and $s = \sqrt[3]{V}$ to find the side.
• Do not confuse cubing ($n^3$) with multiplying by 3 ($3n$).

Return to Pre-Algebra for more topics in this section.