Perfect Cubes — Definition & Examples

Perfect Cube

To understand what a perfect cube is, it's helpful to look at a few shapes. Specifically, the name "perfect cubes" suggest we consider cubes from geometry.

A cube is just a three dimensional square. Every side has the same length. A common example of a cube is a die, like you would use when playing Yahtzee.

If one side of the cube has a length of $5$, then every side of the cube has a length of $5$. Believe it or not, each cube has $12$ sides. It is made out of six squares, which are called faces.

To find the volume of a cube, we multiply the length, width, and height together. According to the order of operations, we multiply from left to right $$\begin{align} V&=l\times w\times h\\ V&=5\times5\times5\\ V&=25\times5\\ V&=125 \end{align}$$

For simplicity we have omitted any units. Some possible options are centimeters, meters, and kilometers.

We refer to $125$ as a perfect cube because it's the product of an integer multiplied by itself three times, $5\times5\times5$. In other words, the cube root of 125 is $5$. A cube root can also be written as a fractional exponent: $\sqrt[3]{125} = 125^{\frac{1}{3}}$.

By multiplying other integers together with themselves, we can find other perfect cubes. $$\begin{align} 1\times1\times1&=1\\ 2\times2\times2&=8\\ 3\times3\times3&=27\\ 4\times4\times4&=64\\ 5\times5\times5&=125\\ 6\times6\times6&=216\\ 7\times7\times7&=343\\ 8\times8\times8&=512\\ 9\times9\times9&=729\\ 10\times10\times10&=1{,}000 \end{align}$$

There are infinitely many perfect cubes. Find the cube of any integer you want, and you've found yourself another perfect cube! For some of the bigger ones, you might want a calculator to find them. However, it's worth memorizing the ten we listed above.

A perfect cube is any number that can be expressed as the product of three equal integers. In other words, a perfect cube is a number that can be made by cubing an integer. Every perfect cube is itself an integer.

Also, the perfect cube will have the same sign as the number you started with.

-The cube of a positive integer is another positive integer

-The cube of a negative integer is another negative integer

-The cube of zero is zero. $\\$

For example, $8$ is a perfect cube because $2^{3}= 2 \times 2 \times 2= 8$ $\\$ Also, $27$ is a perfect cube because $3^3= 3 \times 3 \times 3=27$. $\\$ However, numbers like $35$, $63$, and $121$ are not perfect cubes because you cannot express any of them as products of three equal integers. $\\$ Here are some other perfect cubes. $$\begin{align} 1\times1\times1&=1\\ 2\times2\times2&=8\\ 3\times3\times3&=27\\ 4\times4\times4 &=64\\ 5\times5\times5&=125\\ 6\times6\times6&=216\\ 7\times7\times7&=343\\ 8\times8\times8&=512\\ 9\times9\times9&=729\\ 10\times10\times10&=1000\\ \end{align}$$

Most integers are not perfect cubes. As you move to the right (or the left) on the number line, you'd notice the perfect cubes get more spread out. Let’s try some examples.

$343$ is a perfect cube because it a number that can be expressed as a product of three equal integers. $$343 = 7 \times 7 \times 7= 7^{3}$$

$72$ is not a perfect cube because you cannot express it as a product of three equal integers.

Here's one other way to think about this question. Notice that $64$ and $125$ are perfect cubes. Specifically,

$$4 \times 4 \times 4 = 64$$

but

$$5 \times 5 \times 5 = 125$$

There are no perfect cubes between $64$ and $125$. This is because as the number we cube increases, so does the value of the perfect cube.

$\\$

(Video) What is a Perfect Cube?

by mathwithmrbarnes

This video by mathwithmrbarnes goes over the connection between perfect cubes and geometric cubes.

$\large{\text{Summary}}$

The volume of a cube is defined as the following: $$\text{Volume} = \text{Length} \cdot \text{Width} \cdot \text{Height}$$ A perfect geometric cube occurs when a cube has the same length, height, and width. Let's use the $3$ squares on each side of a Rubix Cube as our unit. $$\begin{align}V&=L \cdot W \cdot H\\ V&= 3 \cdot 3 \cdot 3\\ V&=3^3\\ V&=27 \end{align}$$ We see that $3$ is multiplied by itself $3$ times, which is the same as $3^3$ or $3$ cubed. These are all mathematical expressions which mean the same thing. Hence they are equivalent.

Any of these are equal to $27$, which is a perfect cube. The number $27$ is a perfect cube because it is the product of an integer multiplied $3$ times.

Any integer multiplied by itself $3$ times will make a perfect cube: $$\begin{align} 1\cdot 1\cdot 1&= 1\\ 2\cdot 2 \cdot 2&=8\\ 3\cdot3\cdot3&=27\\ 4\cdot4\cdot4&=64\\ \end{align}$$

This can continue for as large a whole number as you like. You can also find cubes of negative integers, for instance

$$(-5)^3 = (-5) \cdot (-5) \cdot (-5) = -125$$

In this case we are multiplying a negative number by itself. Hence there are lots of minus signs in the equation above.

Perfect cubes can be simplified with a cube root. As a result, the two equations below mean the same thing. You'd use the first if you were given the side length and wanted to know volume. You'd use the second if you were given the volume and wanted to know side length. In this case, if you find one side length, you've found them all. In a cube, all of the side lengths are the same!

$$\begin{align}\text{Volume} &= (\text{Side Length})^3\\ \text{Side Length} &= \sqrt[3]{\text{Volume}} \end{align}$$

Perfect Cubes

Similarly to perfect squares, perfect cubes are the result of multiplying a number by itself $2$ times! In terms of exponents, we say that we cube a number when we take any number to an exponent of $3$. While we can cube any real number, we're going to focus on cubing integers.

Think of drawing a perfect cube. What makes it "perfect"? Well, a "perfect cube" is a three-dimensional figure composed of $6$ perfect squares, which means that every side in the cube is the same size!

Let's look at a list of perfect cubes!

$$\begin{align} 1^3 &= 1 \\ 2^3 &= 8\\ 3^3 &= 27\\ 4^3 &= 64\\ 5^3 &= 125\\ 6^3 &= 216\\ 7^3 &= 343\\ 8^3 &= 512\\ 9^3 &= 729\\ 10^3 &= 1000\\ \end{align}$$

Each of these is multiplying a positive integer by itself twice.

It is worth noting that when we say "perfect cube", we are talking about the numbers to the right of the equal sign above. Also, as the number we are cubing increases, so does the final answer.

For example, we would calculate $3^3$ as $3 \cdot 3 \cdot 3$. Through order of operations, this simplifies to $9 \cdot 3$, which we know is equal to $27$.

So, $27$ is a perfect cube! If I were you I'd memorize the ten perfect cubes above. They come up a lot in other problems. For calculating bigger perfect cubes, you may need a calculator.