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, let’s 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, let’s consider an example. We want to find the number that gives when cubed. To do this, let’s 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
Let’s 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
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 .
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 1 000 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 .
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 cm3.
We first recall that a cube of side length cm will have a volume of cm3. 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 cm3.
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
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 .
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
We then rewrite the fractions to have a common denominator so that we can add them:
Let’s finish by recapping some of the important points from this explainer.
- 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.