In this explainer, we will learn how to find cube roots of rational numbers.

We begin by recalling that the cube root of a perfect cube , written , is the integer such that . In particular, we have .

Finding the cube root of a number is the same as determining the length of a side of a cube from its volume. However, it is possible to find the cube root of 0 or negative numbers, even if this geometric interpretation no longer holds.

To do this, letβs first consider a useful property of the cube root of a number by considering the product of two perfect cubes, say and . We note that

Therefore, when we cube , we get . This means that is the cube root of . We have proven the following result.

### Property: Cube Root of the Product of Perfect Cubes

If and are integers, then

We can use a similar idea to extend this definition of the cube root to take the cube roots of the quotients of integers; these are called rational numbers.

To do this, letβs consider an example. We want to find the number that gives when cubed. To do this, letβs start with a rational number such that and share no common factors. We want

We can see that . So, . We can actually prove a similar result to the product by using a very similar proof. We first consider the cube of for integers and , where is nonzero, as follows

Hence, is the cube root of . This gives us the following result.

### Property: Cube Root of Rational Numbers

If and are integers and , then

Letβs see an example of using this property to determine the cube root of a rational number.

### Example 1: Finding the Cube Root of a Rational Number

Evaluate .

### Answer

We recall that the cube root of a number , written , is the number such that .

In particular, if and are integers and , then .

We note that and . Hence,

In our next example, we will determine the cube root of a rational number given in decimal form.

### Example 2: Finding the Cube Root of a Rational Number in Decimal Form

Find the value of .

### Answer

To determine the cube root of this number, we could start by converting the decimal into a fraction. One way of doing this is to note that the decimal expansion is finite and that

Dividing this equation through by 1βββ000 gives

This means we are asked to find the value of .

We can now recall that if and are integers and , then . We note that and . Thus, and .

Hence,

Since our original value was given as a decimal, we can also give this as the decimal 0.3.

This is not the only method we can use. We can also note that . Using the laws of exponents, we have

In our next example, we will determine the length of the sides of a cube from its volume.

### Example 3: Finding the Side Length of a Cube given Its Volume

Find the side length of a cube given that its volume is cm^{3}.

### Answer

We first recall that a cube of side length cm will have a volume of
cm^{3}. Hence, we must have

We can determine if 27 is a perfect cube by factoring it into primes. We note that 3 is a factor and that . We also note that 3 is a factor of 9, and so . So, . We can also see that . Thus, we have

Finally, since the cube root of is , we must have that .

Hence, the sides of the cube have a length of cm.

In our next example, we will determine the radius of a sphere from its volume.

### Example 4: Finding the Radius of a Sphere given Its Volume

Assuming that the value of is , find the radius of a sphere given its volume is
cm^{3}.

### Answer

We begin by recalling that a sphere of radius will have a volume given by the formula

We should first convert the volume into a fraction. We can do this by noting that

Thus,

Now, by substituting the volume with and with , we get

Dividing the equation through by gives

If we factor 343 into primes, we note that . Similarly, we can factor the denominator into primes to see that . Hence, we have

Since the cube root of is , we must have that .

Writing this as a decimal, we have that the radius is 3.5 cm.

In our next example, we will evaluate an expression involving the cube root of rational numbers.

### Example 5: Evaluating Numerical Expressions Involving Cube Roots with Negative Numbers

Find the value of .

### Answer

We recall that if and are integers and , then

We can note that , , , and . This then allows us to evaluate both cube roots. We have

Hence,

We then rewrite the fractions to have a common denominator so that we can add them:

Letβs finish by recapping some of the important points from this explainer.

### Key Points

- If and are integers, then
- If and are integers and , then .
- We may need to factor the numerator and denominator into primes to determine their cube roots.
- We can use cube roots for some geometric applications, such as determining the side length of a cube or the radius of a sphere.