Lesson Explainer: Cube Roots of Rational Numbers Mathematics

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 𝑛, written οŽ’βˆšπ‘›, is the integer π‘Ž such that π‘Ž=π‘›οŠ©. In particular, we have οŽ’οŽ’βˆšπ‘›=βˆšπ‘Ž=π‘ŽοŠ©.

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 π‘ŽοŠ© and π‘οŠ©. We note that (π‘ŽΓ—π‘)=(π‘ŽΓ—π‘)Γ—(π‘ŽΓ—π‘)Γ—(π‘ŽΓ—π‘)=(π‘ŽΓ—π‘ŽΓ—π‘Ž)Γ—(𝑏×𝑏×𝑏)=π‘Žπ‘.

Therefore, when we cube π‘ŽΓ—π‘, we get π‘Žπ‘οŠ©οŠ©. This means that π‘Žπ‘ is the cube root of π‘Žπ‘οŠ©οŠ©. We have proven the following result.

Property: Cube Root of the Product of Perfect Cubes

If π‘Ž and 𝑏 are integers, then οŽ’βˆšπ‘Žπ‘=π‘Žπ‘.

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 18 when cubed. To do this, let’s start with a rational number π‘Žπ‘ such that π‘Ž and 𝑏 share no common factors. We want ο€»π‘Žπ‘ο‡=18.

We can see that ο€Ό12=12=18. So, οŽ’ο„ž18=12. We can actually prove a similar result to the product by using a very similar proof. We first consider the cube of π‘Žπ‘ for integers π‘Ž and 𝑏, where 𝑏 is nonzero, as follows ο€»π‘Žπ‘ο‡=ο€»π‘Žπ‘ο‡Γ—ο€»π‘Žπ‘ο‡Γ—ο€»π‘Žπ‘ο‡=π‘ŽΓ—π‘ŽΓ—π‘Žπ‘Γ—π‘Γ—π‘=π‘Žπ‘.

Hence, π‘Žπ‘ is the cube root of π‘Žπ‘οŠ©οŠ©. This gives us the following result.

Property: Cube Root of Rational Numbers

If π‘Ž and 𝑏 are integers and 𝑏≠0, then οŽ’οŽ’οŽ’ο„žπ‘Žπ‘=βˆšπ‘Žβˆšπ‘=π‘Žπ‘.

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

Evaluate οŽ’ο„ž64343.

Answer

We recall that the cube root of a number 𝑛, written οŽ’βˆšπ‘›, is the number π‘Ž such that π‘Ž=π‘›οŠ©.

In particular, if 𝑛 and π‘š are integers and π‘šβ‰ 0, then οŽ’ο„žπ‘›π‘š=π‘›π‘šοŠ©οŠ©.

We note that 64=4 and 343=7. Hence, οŽ’οŽ’ο„ž64343=ο„ž47=47.

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 √0.027.

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 0.027Γ—1000=27.

Dividing this equation through by 1β€Žβ€‰β€Ž000 gives 0.027=271000.

This means we are asked to find the value of οŽ’ο„ž271000.

We can now recall that if 𝑛 and π‘š are integers and π‘šβ‰ 0, then οŽ’οŽ’οŽ’ο„žπ‘›π‘š=βˆšπ‘›βˆšπ‘š. We note that 27=3 and 1000=10. Thus, √27=3 and √1000=10.

Hence, √0.027=ο„ž271000=√27√1000=310.

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 0.027=27Γ—0.001=3Γ—0.1. Using the laws of exponents, we have √0.027=√3Γ—0.1=(3Γ—0.1)=3Γ—0.1=0.3.

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 278 cm3.

Answer

We first recall that a cube of side length 𝑙 cm will have a volume of π‘™οŠ© cm3. Hence, we must have 𝑙=278.

We can determine if 27 is a perfect cube by factoring it into primes. We note that 3 is a factor and that 27=3Γ—9. We also note that 3 is a factor of 9, and so 27=3Γ—3Γ—3. So, 27=3. We can also see that 8=2. Thus, we have 𝑙=32=ο€Ό32.

Finally, since the cube root of ο€Ό32 is 32, we must have that 𝑙=32.

Hence, the sides of the cube have a length of 32 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 πœ‹ is 227, find the radius of a sphere given its volume is 179.Μ‡6 cm3.

Answer

We begin by recalling that a sphere of radius π‘Ÿ will have a volume given by the formula volume=43πœ‹π‘Ÿ.

We should first convert the volume into a fraction. We can do this by noting that 0.Μ‡6=23.

Thus, 179.Μ‡6=179+23=179Γ—33+23=5393.

Now, by substituting the volume with 5393 and πœ‹ with 227, we get 5393=43Γ—ο€Ό227οˆπ‘Ÿ=8821π‘Ÿ.

Dividing the equation through by 8821 gives π‘Ÿ=3438.

If we factor 343 into primes, we note that 343=7. Similarly, we can factor the denominator into primes to see that 8=2. Hence, we have π‘Ÿ=72=ο€Ό72.

Since the cube root of ο€Ό72 is 72, we must have that π‘Ÿ=72.

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 οŽ’οŽ’ο„žβˆ’1000729+ο„ž271000.

Answer

We recall that if 𝑛 and π‘š are integers and π‘šβ‰ 0, then οŽ’ο„žπ‘›π‘š=π‘›π‘š.

We can note that 1000=10, βˆ’1000=βˆ’10, 729=9, and 27=3. This then allows us to evaluate both cube roots. We have οŽ’οŽ’οŽ’οŽ’ο„žβˆ’1000729=ο„Ÿ(βˆ’10)9=βˆ’109,ο„ž271000=ο„ž310=310.

Hence, οŽ’οŽ’ο„žβˆ’1000729+ο„ž271000=βˆ’109+310.

We then rewrite the fractions to have a common denominator so that we can add them: βˆ’109+310=βˆ’109Γ—1010+310Γ—99=βˆ’10090+2790=βˆ’7390.

Let’s finish by recapping some of the important points from this explainer.

Key Points

  • If π‘Ž and 𝑏 are integers, then οŽ’βˆšπ‘Žπ‘=π‘Žπ‘.
  • If π‘Ž and 𝑏 are integers and 𝑏≠0, then οŽ’ο„žπ‘Žπ‘=π‘Žπ‘οŠ©οŠ©.
  • 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.

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