CHAPTER
11

Cubed Numbers

In This Chapter

• Learning how to find the cube of any two-digit number
• Finding the cubes of numbers near a base
• Applying the ratio to find the cube of a number

Now that you know how to find squares, let’s move on to cubes. Like with squared numbers, it’s not difficult to put aside the calculator to find the answers once you know some quick and interesting shortcuts.

In this chapter, I talk about finding the value of two-digit cubed numbers and cubed numbers near a base.

Finding the Cubes of Two-Digit Numbers

Do you know the formula for the expansion of (a + b)³? This formula is important, as it will help you find the cubes of various numbers. Let’s see the formula:

In this formula, consider a to be in the tens place and b the ones place. If you take a closer look at the first line, you can spot a ratio between a³ and a²b; if you divide a²b by a³, you get the ratio . Similarly, you get when you divide ab² by a²b and ab² by b³. The ratio is very helpful in calculating the complete value of the cube. We multiply each term in the first line by this ratio.

In the second line, a²b has been doubled to 2a²b and ab² has been doubled to 2ab². Added together, these two lines give you the expression for (a + b)³, which is a³ + 3a²b + 3ab² + b³. That means you can find the numbers for the two lines and combine them to find a two-digit number to the third power.

Take a look at the following examples to see how you can apply the formula to find the cube of a two-digit number.

Example 1

Find the value of 13³.

Step 1: Here, a is 1 and b is 3. Cube a: 1³ = 1. The 1 is the first number in the line.

13³ = 1

Step 2: Find the rest of the numbers for the first line as set up in the expansion of (a + b)³. Because is , multiply each subsequent digit by 3 until you get to the b³ value: 1 × 3 = 3, 3 × 3 = 9, and 9 × 3 = 27. The last number, 27, is equal to b³, so the first line is complete.

Step 3: To get the second line, multiply the second and third terms by 2, as you saw in the expansion: 3 × 2 = 6 and 9 × 2 = 18.

Step 4: Start adding from right to left, making sure every column before the last only has one digit in it. In the first column, bring down the 7 and carry over the 2 to the next step.

Step 5: In the next column, add 9, 18, and the carryover: 9 + 18 + 2 = 29. Put down 9 and carry over the 2 to the next step.

Step 6: In the next column, add 3, 6, and the carryover: 3 + 6 + 2 = 11. Put down 1 and carry over the 1 to the next step.

Step 7: In the last column, add 1 and the carryover: 1 + 1 = 2. Put down 2.

Solution: The answer is 2,197.

QUICK TIP

Remember, when getting cubes, each column before the last should have only one digit. That will make your addition very easy.

Example 2

Find the value of 32³.

Step 1: Here, a is 3 and b is 2. Cube a: 3³ = 27. The 27 is the first number in the line.

32³ = 27

Step 2: Find the rest of the numbers for the first line as set up in the expansion of (a + b)³. Because is , multiply each subsequent digit by 2 and divide by 3 until you get to the b³ value: (27 × 2) ÷ 3 = 18, (18 × 2) ÷ 3 = 12, and (12 × 2) ÷ 3 = 8. The last number, 8, is equal to b³, so the first line is complete.

Step 3: To get the second line, multiply the second and third terms by 2, as you saw in the expansion: 18 × 2 = 36 and 12 × 2 = 24.

Step 4: Start adding from right to left, making sure every column before the last only has one digit in it. In the first column, bring down 8.

Step 5: In the next column, add 12 and 24: 12 + 24 = 36. Put down 6 and carry over the 3 to the next step.

Step 6: In the next column, add 18, 36, and the carryover: 18 + 36 + 3 = 57. Put down 7 and carry over the 5 to the next step.

Step 7: In the last column, add 27 and the carryover: 27 + 5 = 32. Put down 32.

Solution: The answer is 32,768.

Example 3

Find the value of 38³.

Step 1: Here, a is 3 and b is 8. Cube a: 3³ = 27. The 27 is the first number in the line.

38³ = 27

Step 2: Find the rest of the numbers for the first line as set up in the expansion of (a + b)³. Because is , multiply each subsequent digit by 8 and divide by 3 until you get to the b³ value: (27 × 8) ÷ 3 = 72, (78 × 8) ÷ 3 = 192, and (192 × 8) ÷ 3 = 512. The last number, 512, is equal to b³, so the first line is complete.

Step 3: To get the second line, multiply the second and third terms by 2, as you saw in the expansion: 72 × 2 = 144 and 192 × 2 = 384.

Step 4: Start adding from right to left, making sure every column before the last only has one digit in it. In the first column, bring down 2 and carry over the 51 to the next step.

Step 5: In the next column, add 192, 384, and the carryover: 192 + 384 + 51 = 627. Put down 7 and carry over the 62 to the next step.

Step 6: In the next column, add 72, 144, and the carryover: 72 + 144 + 62 = 278. Put down 8 and carry over the 27 to the next step.

Step 7: In the last column, add 27 and the carryover: 27 + 27 = 54. Put down 54.

Solution: The answer is 54,872.

Calculating Cubes Near a Base

If the number being cubed is near a base of 100; 1,000; 10,000; and so on, there’s a process you can use to more easily find the answer. First, you need to know the cubes of numbers 1 through 9, as you’ll be using them in your calculations.

Now that you’ve looked over the cube values, the following sections detail how to find the cube depending on whether your number is above or below the base.

QUICK TIP

At some point, take the time to memorize these values. The cube of a single-digit number is part of the process, so having them at the ready will make that part of solving the problem a breeze.

When the Number Is Above the Base

To find the cube of a number above the base, you add the number to two times its excess of the base to get the first part of your answer. You then square the excess and multiply it by 3 to get the second part. To get the third and final part, you find the cube of the excess.

Example 1

Find the value of 105³.

Step 1: This number is closest to a base of 100; the excess is 5, as 105 is 5 more than the base. To get the first part of the answer, add 105 to two times the excess: 105 + (5 × 2) = 115.

105³ = 115

Step 2: To get the second part of the answer, square the excess: 5² = 25. Multiply that number by 3: 25 × 3 = 75.

105³ = 115/75

Step 3: To find the third and final part of the answer, find the cube of the excess (see the earlier table if you need a refresher): 5³ = 125.

105³ = 115/75/125

Step 4: Combine the three parts. The second and third part of the answer should only contain the number of digits equal to the number of zeroes in the base. In this case, because the base is 100, there should only be two digits in those parts. That means, for the third part, you need to carry over the 1 and add it to the second part: 75 + 1 = 76.

105³ = 115/75/125

105³ = 115/76/25 = 1157625

Solution: The answer is 1,157,625.

Example 2

Find the value of 1,009³.

Step 1: This number is closest to a base of 1,000; the excess is 9, as 1,009 is 9 more than the base. To get the first part of the answer, add 1,009 to two times the excess: 1,009 + (9 × 2) = 1,027.

1009³ = 1027

Step 2: To get the second part of the answer, square the excess: 9² = 81. Multiply that number by 3: 81 × 3 = 243.

1009³ = 1027/243

Step 3: To find the third and final part of the answer, find the cube of the excess (see the earlier table if you need a refresher): 9³ = 729.

1009³ = 1027/243/729

Step 4: Combine the three parts. The second and third part of the answer should only contain the number of digits equal to the number of zeroes in the base. In this case, because the base is 1,000, there should only be three digits in those parts. Both parts already have three digits, so you don’t need to do any carryovers.

1009³ = 1027/243/729 = 1027243729

Solution: The answer is 1,027,243,729.

Example 3

Find the value of 10,006³.

Step 1: This number is closest to a base of 10,000; the excess is 6, as 10,009 is 6 more than the base. To get the first part of the answer, add 10,006 to two times the excess: 10,006 + (6 × 2) 10,018.

10006³ = 10018

Step 2: To get the second part of the answer, square the excess: 6² = 36. Multiply that number by 3: 36 × 3 = 108.

10006³ = 10018/108

Step 3: To find the third and final part of the answer, find the cube of the excess (see the earlier table if you need a refresher): 6³ = 216.

10006³ = 10018/108/216

Step 4: Combine the three parts. The second and third part of the answer should only contain the number of digits equal to the number of zeroes in the base. Because the base is 10,000, there should be four digits in those parts. In this case, you need to add zeroes before the 108 and 216 to make them four digits.

10006³ = 10018/108/216

10006³ = 10018/0108/0216 = 1001801080216

Solution: The answer is 1,001,801,080,216.

When the Number Is Below the Base

For a number below the base, you have to change up the process a bit. The deficit is treated as a negative number, so instead of adding in the first part, you subtract. The second part is exactly the same process—you square the difference and multiply it by 3. The process for getting the third part is the same as well—you cube the difference—but what you do with it when combining the parts is altered. The third part of the answer will be a negative number, so in order to get rid of the negative, you carry over the base from the second part and subtract the third part from that.

Example 1

Find the value of 96³.

Step 1: This number is closest to a base of 100; the deficit is 4, as 96 is 4 less than the base. To get the first part of the answer, subtract 96 by two times the deficit: 96 – (4 × 2) = 88.

96³ = 88

Step 2: To get the second part of the answer, square the deficit: −4² = 16. Multiply that number by 3: 16 × 3 = 48.

96³ = 88/48

Step 3: To find the third and final part of the answer, find the cube of the excess (see the earlier table if you need a refresher; this version has a negative): −4³ = −64.

96³ = 88/48/-64

Step 4: Combine the three parts. You can’t have a negative number in the answer, so for the third part, carry over base 100 from the second part. That simply means subtracting 1 from 48, as 8 is in the hundreds place: 48 − 1 = 47. You then subtract the third part from 100: 100 − 64 = 36.

96³ = 88/48/-64

96³ = 88/47/36 = 884736

Solution: The answer is 884,736.

Example 2

Find the value of 992³.

Step 1: This number is closest to a base of 1,000; the deficit is 8, as 992 is 8 less than the base. To get the first part of the answer, subtract 992 by two times the deficit: 992 – (8 × 2) = 976.

992³ = 976

Step 2: To get the second part of the answer, square the deficit: −8² = 64. Multiply that number by 3: 64 × 3 = 192.

992³ = 976/192

Step 3: To find the third and final part of the answer, find the cube of the excess (see the earlier table if you need a refresher; this version has a negative): −8³= −512.

992³ = 976/192/-512

Step 4: Combine the three parts. You can’t have a negative number in the answer, so for the third part, carry over base 1,000 from the second part. That simply means subtracting 1 from 192, as 2 is in the thousands place: 192 − 1 = 191. You then subtract the third part from 1,000: 1,000 − 512 = 488.

992³ = 976/192/-512

992³ = 976/191/488 = 976191488

Solution: The answer is 976,191,488.

Example 3

Find the value of 9,993³.

Step 1: This number is closest to a base of 10,000; the deficit is 7, as 9,993 is 7 less than the base. To get the first part of the answer, subtract 9,993 by two times the deficit: 9,993 − 14 = 9,979.

9993³ = 9979

Step 2: To get the second part of the answer, square the deficit: -7² = 49. Multiply that number by 3: 49 × 3 = 147. Write this as 0147 because of the placement rule.

9993³ = 9979/0147

QUICK TIP

The placement rule states that the number of digits in the column or section should match the number of zeroes in the base. For base 10,000, there are four zeroes; therefore, 147 is written as 0147 to make it a four-digit number.

Step 3: To find the third and final part of the answer, find the cube of the excess (see the earlier table if you need a refresher; this version has a negative): −7³ = −343. Write this as −0343 because of the placement rule.

9993³ = 9979/0147/-0343

Step 4: Combine the three parts. You can’t have a negative number in the answer, so for the third part, carry over base 10,000 from the second part. That simply means subtracting 1 from 0147, as 7 is in the ten thousands place: 0147 − 1 = 0146. You then subtract the third part from 10,000: 10,000 − 0343 = 9,657.

9993³ = 9979/0147/-0343

9993³ = 9979/0146/9657 = 997901469657

Solution: The answer is 997,901,469,657.

The Least You Need to Know

• To find out how much a number is cubed, use a³ + a²b + a ² b² + ab ² + b³ and 2a²b + 2ab² to find the values, and add right to left.
• Use to find the numbers for the first line of the formula.
• If the number being cubed is above or below a base, you can use the excess or deficit to help you find the value.