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Finding the Cube Root of Decimals

By Kathleen Knowles, 18 Jul 2020

A decimal number is essentially a fraction that uses a point to separate the whole number part and the fractional part. The fractional parts are expressed as parts in tens, hundreds, thousands, etc. An example of a decimal number is 2.5, where 2 is the whole number, and 5 is the fractional part. The 5 after the decimal point means 5/10.

Another example of a decimal number is 23.025. In this example, 23 is the whole number, while 025 is the fraction. In this instance, the 025 means 25/1000 or 25 parts out of 1000. As a rule, the number of digits after the decimal place denotes the number of zeroes used in the denominator of the fraction. So one zero would indicate tenths, while two indicates hundredths.

The cube root of a decimal number is a number that, when multiplied by itself three times, will give the result in the decimal number. For example, the cube root of 8 would be 2 because 2 * 2 * 2 = 8.

Perfect Cubes

Perfect cubes are numbers whose cube roots are whole numbers. Eight is a perfect cube because its cube root, 2, is a whole number. Examples of perfect cubes are as follows:

  • 23 = 8
  • 33 = 27
  • 43 = 64
  • 53 = 125
  • 63 = 216
  • 73 = 343
  • 83 = 512
  • 93 = 729
  • 103 = 1000

Cube Root Of Numbers Whole Numbers

Let’s start with the cube roots of non-fractional decimal numbers.

Example 1:

Find the cube root of 216.0.

The number 216 is a whole number. The first step is to express the number as a product of its prime factors. When the number is expressed as a product of prime factors, the factors can then be grouped, and the cube root picked from the groups. Let’s see how this works.

Step 1

The first number to start with is 2. It's the smallest prime number. Let’s check if 2 is a factor of 216. If the last digit of an integer is either zero or an even number, then the integer is divisible by 2: 216 / 2 = 108.

Our example, 216, ends with 6, which is even. So 2 is a prime factor. The result, 108, is also divisible by 2: 108 / 2 = 54. Also, 54 is divisible by 2: 54 / 2 = 27.

Notice that 27 ends with 7, which is not even, so it is not divisible by 2. However, 27 is divisible by 3: 27 / 3 = 9. Then, 9 is also divisible by 3: 9 / 3 = 3. And 3 is also divisible by 3: 3 / 3 = 1. We stop the division when we arrive at 1.

Step 2

Express the number whose cube root you intend to find as a product of prime factors:

216 = 2 * 2 * 2 * 3 * 3 * 3

Step 3

Pick the prime factors that occur up to three times. From the above prime factors, 2 appears three times, and 3 appears three times, so we pick 2 and 3:

2 * 3 = 6

The product of the numbers we have picked is 6, so the cube root of 216 is 6.

Example 2

Find the cube root of 3375. We will follow the same steps as in the above example.

Step 1

Express 3375 as a product of prime factors. 3375 ends with 5, which is an odd number, so 2 is not a factor of 3375. Let us check for 3. To check if 3 is a factor of a number, sum the digits of the number and divide the sum by 3. If the remainder is zero, then 3 is a factor of the number. The sum of the digits of 3375 is 18. 18 / 3 is 6. So 3 is a factor of 3375: 3375 / 3 = 1125

Using the rule we established earlier, 1125 is divisible by 3: 1125 / 3 = 375

375 is also divisible by 3: 375 / 3 = 125.

But 125 is no longer divisible by 3. Let’s check the next prime number, 5. All integers that end with 5 or 0 are divisible by 5: 125 / 5 = 25

25 is also divisible by 5: 25 / 5 = 5. And 5 is also divisible by 5: 5 / 5 = 1. Since the result is now 1, we will stop the division.

Step 2

The next step is to express 3375 as a product of prime factors

3375 = 3 * 3 * 3 * 5 * 5 * 5

Step 3

We will now pick the prime factors that appear 3 times. 3 and 5 have appeared 3 times each.

3 * 5 = 15

The cube root of 3375 is 15.

Cube Root of Decimals with a Fractional Part

This refers to numbers that are not whole numbers and are expressed in decimal fractions.

Example 3

What is the cube root of 0.343?

0.343 has a whole number part, which is 0, and the fractional part, which is 343.

Step 1

The first step is to express the number as a fraction with a numerator and a denominator.

0.343 = 343 / 1000

The numerator is 343, and the denominator is 1000

Step 2

The next step is to find the cube root of both the numerator and the denominator using the method of factorization explained in the examples above.

Let's look at the prime factors of 343, which is not divisible by 2, 3, or 5. It is divisible by 7

  • 343 / 7 = 49
  • 49 / 7 = 7
  • 7 / 7 = 1
  • 343 = 7 * 7 * 7

For the denominator, we already know that 1000 = 10 * 10 * 10

Step 3

Pick the factors that occur three times: 7 for the numerator and 10 for the denominator. Hence the cube root of 343 / 1000 = 7 / 10. The cube root of 0.343 is the same as the cube root of 343/1000 = 7/10 = 0.7.

So the cube root of 0.343 = 0.7.

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