Lesson Explainer: Operations on Cube Roots | Nagwa Lesson Explainer: Operations on Cube Roots | Nagwa

Lesson Explainer: Operations on Cube Roots Mathematics

In this explainer, we will learn how to use the properties of operations on cube roots to simplify expressions.

We begin by recalling that, for any two real numbers 𝑎 and 𝑏, we have that 𝑎×𝑏=𝑎×𝑏. We can prove this fact by noting that 𝑎×𝑏=𝑎×𝑏×𝑎×𝑏×𝑎×𝑏=𝑎×𝑏=𝑎×𝑏.

We can write this formally as follows.

Property: The Product of Cube Roots

For any real numbers 𝑎 and 𝑏, we have that 𝑎×𝑏=𝑎×𝑏.

We can use this result in both directions to simplify radical expressions.

First, we can simplify the product of radical expressions. Consider the following example: 12×18=12×18=216=6.

Second, we can simplify a cube root that is divisible by a perfect cube by splitting it into a product of cube roots. For example, 54=27×2=27×2=32.

By using the second method, we can rewrite the cube root of a non-perfect cube in a simplified form. Consider 𝑐 for an integer 𝑐; we can find the greatest perfect cube that divides 𝑐, for example, 𝑎. Then, we have that 𝑐=𝑎×𝑏=𝑎𝑏, where 𝑏=𝑐𝑎.

We want the sign of 𝑎 to represent the sign of the number, so we choose our perfect cube to make 𝑎 the same sign as 𝑐. This gives us the following method for simplifying the cube root of a non-perfect cube.

How To: Simplifying the Cube Root of a Non-Perfect Cube

Given any integer 𝑐, we write 𝑐 in the form 𝑎𝑏, where 𝑏 is the smallest possible positive integer, by using the following steps:

  1. Find the greatest perfect cube that divides 𝑐.
  2. Choose integer 𝑎 to have the same sign as 𝑐 such that 𝑎 divides 𝑐 and 𝑎 is the greatest perfect cube dividing 𝑐.
  3. We then have that 𝑐=𝑎×𝑏=𝑎𝑏.

It is worth noting that we do not immediately need to identify the greatest perfect cube dividing the radicand of the cube root (the number inside the cube root) straight away. This is because we can apply this process multiple times. For example, 1080=8×135=2135=227×5=235=65.

Let’s now see an example of applying this process to simplify the cube root of integer values.

Example 1: Expressing a Cube Root in Its Simplest Form

Write each of the following radical expressions in the form 𝑎𝑏, where 𝑎 and 𝑏 are integers and 𝑏 is the smallest possible positive value. 256540

Answer

Part 1

We start by looking for perfect cubes that divide 256. We see that 2=8 is a factor of 256 since 256=8×32, and we recall that, for any two real numbers 𝑛 and 𝑚, we have that 𝑛×𝑚=𝑛×𝑚. Hence, 256=8×32=8×32=232.

We might be tempted to stop here; however, we need 𝑏 to be the smallest possible positive value such that no perfect cubes can divide 𝑏. We see that 32=8×4, so we can apply this process again. We have 232=28×4=28×4=224=44.

By inspection, we can see that there are no more perfect cubes that divide 4, so we cannot simplify further. Hence, 256=44.

Part 2

We want to apply the same process; however, this time, since the radicand of the cube root is negative, we need to find a negative perfect cube divisor of 540. We note that 540=(27)×20. Hence, 540=(27)×20=27×20=320.

By inspection, we can see that there are no more perfect cubes that divide 20, so we cannot simplify further. Hence, 540=320.

In our next example, we will apply the property of the product of cube roots to simplify a product of radicals.

Example 2: Simplifying the Product of Two Cube Roots

Express 4×16 in its simplest form.

Answer

We first recall that, for any two real numbers 𝑎 and 𝑏, we have that 𝑎×𝑏=𝑎×𝑏. Applying this, we have 4×16=4×(16)=64.

We then note that (4)=64; hence, 64=4.

Thus far, we have worked with simplifying the cube roots of integers. However, this result works with the cube roots of any real numbers. An application of this result is to consider the cube root of the quotient of real numbers. We have 𝑎𝑏=𝑎×1𝑏=𝑎×1𝑏.

We note that 1𝑏=1𝑏×1𝑏×1𝑏=1𝑏. Hence, 𝑎×1𝑏=𝑎×1𝑏=𝑎𝑏.

Of course, we assume that 𝑏0 since we are dividing by 𝑏. This gives us the following result.

Property: Cube Root of the Quotient of Real Numbers

For any real numbers 𝑎 and 𝑏, where 𝑏0, we have that 𝑎𝑏=𝑎𝑏.

Now, let’s look at an example of using both properties to simplify an expression involving radicals.

Example 3: Simplifying the Multiplication and Division of Multiple Cube Roots

Simplify 2×432×2.

Answer

We start by noting that none of the radicands are perfect cubes, so we cannot directly evaluate any of the individual radicals. Instead, we will combine the radicals and then simplify. We first recall that, for any real numbers 𝑎 and 𝑏, we have that 𝑎×𝑏=𝑎×𝑏. Applying this, we have 2×432×2=2×432×(2)=864.

We can simplify this expression by prime factoring the radicand of each cube root. We note that 8=2 and 64=(4). So, 864=2(4)=24=12.

In our next example, we will see how this process can be used to simplify the difference between the cube roots of integers.

Example 4: Simplifying the Difference of Two Cube Roots with Different Bases

Express 2564 in its simplest form.

Answer

We first note that neither radicand of the cube roots is a perfect cube, so we cannot evaluate either term directly. Instead, let’s simplify each term by writing them in the form 𝑎𝑏, where 𝑎 and 𝑏 are integers and 𝑏 is the smallest possible positive integer.

Now, we look for perfect cubes that divide 256; we find that 256=8×32. We can then use the fact that, for any real numbers 𝑎 and 𝑏, we have that 𝑎×𝑏=𝑎×𝑏 to simplify as follows: 256=8×32=8×32=232.

We can simplify this further by noting that 32=8×4. Thus, 232=28×4=28×4=224=44.

Substituting this value into the given expression yields 2564=444.

We then take out a factor of 4 to get 444=(41)4=34.

We cannot find a perfect cube divisor of 4, which is greater than 1, so we cannot simplify the radical any further. Hence, 2564=34.

In our next example, we will simplify an expression involving the addition of multiple radical expressions.

Example 5: Simplifying the Addition of Cube Roots

Express 5192+5648+375 in its simplest form.

Answer

We will start by simplifying each term separately; we can do this by using the fact that, for any real numbers 𝑎 and 𝑏, we have that 𝑎×𝑏=𝑎×𝑏.

To do this, we check 192 for any perfect cube divisors. We see that 8 is a factor of 192 since 192=8×24. Hence, 5192=58×24=58×24=5224=1024.

We want the radicand of the cube root to be the smallest positive integer possible. We can do this by checking to see if 24 has any perfect cube divisors. We note that 24=8×3. Thus, 1024=108×3=108×3=102×3=203.

Now, 3 has no perfect cube divisors greater than 1, so we cannot simplify this term any further.

Next, we check 648 for any perfect cube divisors. Since this number is negative, we want a negative perfect cube. We note that 648=(8)×81. Hence, 5648=58×81=5281=1081.

We want the radicand of the cube root to be the smallest positive integer possible. We can do this by checking to see if 81 has any perfect cube divisors. We note that 81=27×3. Thus, 1081=1027×3=1033=303.

Finally, we check 375 for any perfect cube divisors. We see that 375=125×3. Thus, 375=125×3=53.

Substituting these values into the given expression yields 5192+5648+375=203303+53.

We can then take out the shared factor of 3 to get 203303+53=(2030+5)3=453.

In our final example, we will evaluate an algebraic expression by using the properties of the cube root.

Example 6: Evaluating an Algebraic Expression Involving Cube Roots

If 𝑥=375 and 𝑦=81, find the value of (𝑥+𝑦).

Answer

We start by noting that we cannot directly add these radicals together, so instead, we look to simplify each cube root separately.

We need to identify a perfect cube that divides 375; we see that 375=125×3. Hence, by using the fact that 𝑎×𝑏=𝑎×𝑏 for any real numbers 𝑎 and 𝑏, we have 375=125×3=53.

Now, we look for a perfect cube that divides 81; we see that 81=27×3. Hence, 81=27×3=33.

We can then substitute 𝑥=53 and 𝑦=33 into the expression to get (𝑥+𝑦)=53+33.

We can take out the shared factor of 3 to get 53+33=(5+3)3=83.

Finally, we evaluate the exponent as follows: 83=83×83×83=(8×8×8)×3×3×3=512×3=1536.

In the previous example, we may have been tempted to add 375 and 81 by adding their radicands to get 375+81. However, we cannot add radical expressions in this manner. An easy way to see this is to consider 8+8. We know that 8=2, so 8+8=2+2=4.

If we added the radicands, then we would get 16, which is not equal to 4.

In a similar way, we can show that (𝑥+𝑦) is not the same as 𝑥+𝑦. So, we cannot just evaluate the expression by adding the radicands: 375+81375+81.

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

Key Points

  • For any two real numbers 𝑎 and 𝑏, we have that 𝑎×𝑏=𝑎×𝑏.
  • For any two real numbers 𝑎 and 𝑏, where 𝑏0, we have that 𝑎𝑏=𝑎𝑏.
  • If 𝑐 is an integer, we can use these results to write 𝑐=𝑎𝑏, where 𝑏 is smallest positive integer and has no perfect cube divisors greater than 1. This is called the simplest form of 𝑐.

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