A decimal number is essentially a fraction that uses a point to separate\u00a0the 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.<\/p>\n
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\u00a0the denominator of the fraction. So one\u00a0zero\u00a0would indicate tenths, while two\u00a0indicates hundredths.<\/p>\n
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\u00a02 because\u00a02 * 2 * 2 = 8.<\/p>\n
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:<\/p>\n
Let\u2019s start with the cube roots of non-fractional decimal numbers.<\/p>\n
Find the cube root of 216.0.<\/p>\n
The number 216 is a whole number.\u00a0The 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\u2019s see how this works.<\/p>\n
The first number to start with is 2. It's the smallest prime number. Let\u2019s check if 2 is a factor of 216. If the\u00a0last digit of an integer is either zero or an\u00a0even number, then the integer\u00a0is divisible by 2:\u00a0216 \/ 2 = 108.<\/p>\n
Our example, 216, ends with 6, which is even. So 2 is a prime factor. The result, 108, is also divisible by 2:\u00a0108 \/ 2 = 54. Also, 54 is divisible by 2:\u00a054 \/ 2 = 27.<\/p>\n
Notice that 27 ends with 7, which is not even, so it is not divisible by 2. However, 27 is divisible by 3:\u00a027 \/ 3 = 9. Then, 9 is also divisible by 3: 9 \/ 3 = 3.\u00a0And 3 is also divisible by 3:\u00a03 \/ 3 = 1.\u00a0We stop the division when we arrive at 1.<\/p>\n
Express the number whose cube root you intend to find as a product of prime factors:<\/p>\n
216 = 2 * 2 * 2 * 3 * 3 * 3<\/p>\n
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:<\/p>\n
2 * 3 = 6<\/p>\n
The product of the numbers we have picked is 6, so the cube root of 216 is 6.<\/p>\n
Find the cube root of 3375. We will follow the same steps as in the above example.<\/p>\n
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<\/p>\n
Using the rule we established earlier, 1125 is divisible by 3:\u00a01125 \/ 3 = 375<\/p>\n
375 is also divisible by 3:\u00a0375 \/ 3 = 125.<\/p>\n
But 125 is no longer divisible by 3. Let\u2019s check the next prime number, 5. All integers that end with 5 or 0 are divisible by 5: 125 \/ 5 = 25<\/p>\n
25 is also divisible by 5: 25 \/ 5 = 5.\u00a0And 5 is also divisible by 5: 5 \/ 5 = 1.\u00a0Since the result is now 1, we will stop the division.<\/p>\n
The next step is to express 3375 as a product of prime factors<\/p>\n
3375 = 3 * 3 * 3 * 5 * 5 * 5<\/p>\n
We will now pick the prime factors that appear 3 times. 3 and 5 have appeared 3 times each.<\/p>\n
3 * 5 = 15<\/p>\n
The cube root of 3375 is 15.<\/p>\n
This refers to numbers that are not whole numbers and are expressed in decimal fractions.<\/p>\n
What is the cube root of 0.343?<\/p>\n
0.343 has a whole number part, which is 0, and the fractional part, which is 343.<\/p>\n
The first step is to express the number as a fraction with a numerator and a denominator.<\/p>\n
0.343 = 343 \/ 1000<\/p>\n
The numerator is 343, and the denominator is 1000<\/p>\n
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.<\/p>\n
Let's look at the prime factors of 343, which\u00a0is not divisible by 2, 3, or 5. It is divisible by 7<\/p>\n
For the denominator, we already know that 1000 = 10 * 10 * 10<\/p>\n
Pick the factors that occur three times: 7 for the numerator and 10 for the denominator.\u00a0Hence the cube root of 343 \/ 1000 = 7 \/ 10.\u00a0The cube root of 0.343 is the same as the cube root of 343\/1000 = 7\/10 = 0.7.<\/p>\n
So the cube root of 0.343 = 0.7.<\/p>\n