CHAPTER
13

Square and Cube Roots

In This Chapter

• Finding the square root of perfect squares
• Getting the square root of any given number
• Discovering the value of perfect cubes

Seeing the square root symbol on a page can inspire nervousness. In some cases, it can be a guessing game, just trying to multiply a number by itself to see if it’s even close. And then there’s the old standby—the calculator. But finding square or even cube roots doesn’t have to be so frustrating.

In this chapter, you learn some ways to painlessly find the square and cube roots of numbers.

Square Roots of Perfect Squares

To begin the discussion of roots, let me show you how to find square roots for perfect squares. A perfect square is a number whose square root is an integer, or whole number. So for the examples in this section, all of your answers should be whole numbers. Not too hard, right?

Take a look at the following table to see what constitutes a perfect square.

You should notice two things about the squares for these numbers:

1. They have a digit sum (see Chapter 5) of 1, 4, 7, or 9.
2. They end in 1, 4, 5, 6, or 9.

So right away, you have two indicators that can help you determine whether a number is a perfect square.

To find the value of a perfect square, you split the number into groups of two to see how many digits are in the answer. You then look at the early digits to see which square value they’re between, giving you the first part of the answer. The table can help you with the last couple steps, which involve knowing the number that has the same last digit when squared and finding the digit sums of the number and the possible answers.

Example 1

Find .

Step 1: Split the number into groups of two to determine how many digits are in the answer. Because you get two groups, 32 and 49, the number of digits in the answer is going to be two.

Step 2: Look at the first two digits of the number together and figure out which square values it’s closest to. In this case, 32 is more than 25 (5²) and less than 36 (6²). This group is the tens place of your answer, so the first part of the answer is between 50 and 60.

Step 3: Look at the last digit of 3,249. If you recall from the table, 3 and 7 have a last digit of 9 when squared. Therefore, the square is either 53 or 57.

Step 4: Compare the digit sums of 53 and 57 squared to the digit sum of 3,249 to see which one matches.

3249: 3 + 2 + 4 + 9 = 18; 1 + 8 = 9

53²: (5 + 3)² = 8² = 64; 6 + 4 = 10; 1 + 0 = 1

57²: (5 + 7)² = 12² = 144; 1 + 4 + 4 = 9

Solution: The answer is 57.

Example 2

Find .

Step 1: Split the number into groups of two to determine how many digits are in the answer. You get two groups, 96 and 04, so the answer’s going to be two digits.

Step 2: Look at the first two digits of the number together and figure out which square values it’s closest to. In this case, 96 is more than 81 (9²) and less than 100 (10²). This group is the tens place of your answer, so the first part of the answer is between 90 and 100.

Step 3: Look at the last digit of 9,604. If you recall from the table, 2 and 8 have a last digit of 4 when squared. Therefore, the square is either 92 or 98.

Step 4: Compare the digit sums of 92 and 98 squared to the digit sum of 9,604 to see which one matches.

9604: 9 + 6 + 0 + 4 = 19; 1 + 9 = 10; 1 + 0 = 1

92²: (9 + 2)² = 11² = 121; 1 + 2 + 1 = 5

98²: (9 + 8)² = 17² = 289; 2 + 8 + 9 = 19; 1 + 9 = 10;
  1 + 0 = 1

Solution: The answer is 98.

QUICK TIP

What do you do when the digit sum for the potential answers is the same? If this happens, simply find the square of a number between them and see how it compares. For example, the square root of 2,401 is either 41 and 49, both of which have a digit sum of 7. To get the right answer, find the value of 45². Its value is 2,025, which is lower than the original number. That means the square root of 2,401 is 49.

Example 3

Find .

Step 1: Split the number into groups of two to determine how many digits are in the answer. For this number, you have five digits, so get two groups of two and one group of one, meaning there are going to be three digits in the answer.

Step 2: Look at the first three digits of the number together and figure out which square values it’s closest to. In this case, 249 is more than 225 (15²) and less than 256 (16²), so the first two digits of the answer must be 15.

Step 3: Look at the last digit of 24,964. If you recall from the table, 2 and 8 have a last digit of 4 when squared. Therefore, the square is either 152 or 158.

Step 4: Compare the digit sums of 152 and 158 squared to the digit sum of 24,964 to see which one matches.

24964: 2 + 4 + 9 + 6 + 4 = 25; 2 + 5 = 7

152²: (1 + 5 + 2)² = 8² = 64; 6 + 4 = 10; 1 + 0 = 1

158²: (1 + 5 + 8)² = 14² = 196; 1 + 9 + 6 = 16; 1 + 6 = 7

Solution: The answer is 158.

Using Duplexes to Find Square Roots

You can find the square root of any number by cycling through two steps:

1. Divide by 2 times the first digit of your answer.
2. Subtract by the duplexes (see Chapter 10 if you need to jog your memory).

Let me show you how it’s done with some examples.

Example 1

Find .

Step 1: Divide the number into groups of two from right to left. Because 529 is a three-digit number, you have one group of two (29) and one group of one (5). There are two groups, so the answer is going to have two digits before the decimal.

Step 2: For the first group, 5, find the perfect square just less than it. The closest perfect square that’s less than 5 is 4. The square root of 4 is the first digit of your answer, so put down 2. To get the divisor, double 2: 2 × 2 = 4. Put the 4 on the left.

Step 3: Subtract 5 minus the square of 2. So we get 5 − 4 = 1. This 1 is the remainder which is then prefixed to 2, making it 12.

Step 4: Divide 12 by 4: 12 ÷ 4 = 3, remainder 0. Put down 3 and prefix the remainder to the 9 so it becomes 09.

Step 5: Subtract 09 by the duplex of 3. In this case, the duplex of 3 is 3², or 9: 09 − 9 = 0. Dividing 0 by 4 gives you 0, so you’re finished. Because the answer only has two digits, the 0 goes after the decimal.

Solution: The answer is 23.0.

Example 2

Find .

Step 1: Divide the number into groups of two from right to left. Because there are two groups, 46 and 24, the answer is going to contain two digits before the decimal.

Step 2: For the first group, 46, find the perfect square just less than it. The closest perfect square that’s less than 46 is 36. The square root of 36 is the first digit of your answer, so put down 6. To get the divisor, double 6: 6 × 2 = 12. Put the 12 on the left.

Step 3: Subtract 46 minus the square of 6. The square of 6 is 36. So we have 46 − 36 = 10. This 10 is our remainder. We prefix 10 to 2 so it becomes 102.

Step 4: Divide 102 by 12: 102 ÷ 12 = 8, remainder 6. Put down 8 and prefix the remainder to the 4 so it becomes 64.

Step 5: Subtract 64 by the duplex of 8. In this case, the duplex of 8 is 8², or 64: 64 − 64 = 0. Dividing 0 by 12 gives you 0, so you’re finished. Because the answer only has two digits, the 0 goes after the decimal.

Solution: The answer is 68.0.

Example 3

Find .

Step 1: Divide the number into groups of two from right to left. Because there are two groups, 26 and 56, the answer is going to contain two digits before the decimal.

Step 2: For the first group, 26, find the perfect square just less than it. The closest perfect square that’s less than 26 is 25. The square root of 25 is the first digit of your answer, so put down 5. To get the divisor, double 5: 5 × 2 = 10. Put the 10 on the left.

Step 3: Subtract 26 minus the square of 5. The square of 5 is 25. So we have 26 − 25 = 1. This 1 is our remainder which we prefix to 5, making it 15.

Step 4: Divide 15 by 10: 15 ÷ 10 = 1, remainder 5. Put down 1 and prefix the remainder to 6 so it becomes 56.

Step 5: Subtract 56 by the duplex of 1. In this case, the duplex of 1 is 1², or 1: 56 − 1 = 55. To get the number after the decimal, divide 55 by 10: 55 ÷ 10 = 5, remainder 5. Put down 5 after the decimal.

Solution: The answer is 51.5.

Cube Roots of Perfect Cubes

Perfect cubes are numbers whose cube roots are whole numbers. Like you did for perfect squares, you first need to check out the link between numbers 1 through 9 and their cubes.

Here are the four connections you should notice:

1. The digit sums (see Chapter 5) go in a cycle of 1, 8, 9. The digit sum is 1 for 1, 4, and 7; 8 for 2, 5, and 8; and 9 for 3, 6, and 9.
2. If the cube ends in a 1, 4, 5, 6, or 9, its cube root always ends with that same number.
3. If the cube ends in a 8, the cube root ends in a 2; if the cube root ends in a 2, the cube ends in a 8.
4. If the cube ends in a 7, the cube root ends in a 3; if the cube ends in a 3, the cube root ends in a 7.

This information will help you find the cube roots.

SPEED BUMP

Just because the number has a digit sum of 1, 8, or 9 doesn’t necessarily mean it’s a perfect cube; other cubes can have those digit sums. Keep that mind if you encounter numbers for which you have to find the cube root.

To find the cube roots of perfect cubes, you first split the number into groups of three to find out how many digits are in the answer. You then find the first part of the answer using the first group you made. The table can help you with the steps that involve finding the last digit of cube root based on the last digit of the cube.

Example 1

Find the cube root of a perfect cube 3,375.

Step 1: Split the number into groups of three from right to left to determine how many digits are in the answer. There are two groups, 3 and 375, so the answer is going to have two digits.

Step 2: Use the last digit of 3,375 to find the last digit of your answer. The last digit is 5, so according to the table, the last digit of the answer is 5.

Step 3: Look at the first group, 3, and figure out which cube value is less than or equal to it to get the first digit of your answer. The closest cube to 3 is 1. The cube root of 1 is 1, so 1 is the first digit.

Step 4: Compare the digit sums of the number and the answer to be sure they match.

3375: 3 + 3 + 7 + 5 =18; 1 + 8 = 9

15³: (1 + 5)³ = 6³ = 216; 2 + 1 + 6 = 9

Solution: The answer is 15.

Example 2

Find the cube root of the perfect cube 328,509.

Step 1: Split the number into groups of three from right to left to determine how many digits are in the answer. There are two groups, 328 and 509, so the answer is going to have two digits.

Step 2: Use the last digit of 328,509 to find the last digit of your answer. The last digit is 9, so according to the table, the last digit of the answer is 9.

Step 3: Look at the first group, 328, and figure out which cube value is less than or equal to it to get the first digit of your answer. As you can see via the table, the closest cube to 328 is 216. The cube root of 216 is 6, so 6 is the first digit.

Step 4: Compare the digit sums of the number and the answer to be sure they match.

328509: 3 + 2 + 8 + 5 + 0 + 9 = 27; 2 + 7 = 9

69³: (6 + 9)³ = 15³ = 3375; 3 + 3 + 7 + 5 = 18; 1 + 8 = 9

Solution: The answer is 69.

Example 3

Find the cube root of the perfect cube 175,616.

Step 1: Split the number into groups of three from right to left to determine how many digits are in the answer. There are two groups, 175 and 616, so the answer is going to have two digits.

Step 2: Use the last digit of 175,616 to find the last digit of your answer. The last digit is 6, so according to the table, the last digit of the answer is 6.

Step 3: Look at the first group, 175, and figure out which cube value is less than or equal to it to get the first digit of your answer. As you can see via the table, the closest cube to 175 is 125. The cube root of 125 is 5, so 5 is the first digit.

Step 4: Compare of the number and the answer to be sure they match.

175616: 1 + 7 + 5 + 6 + 1 + 6 = 26; 2 + 6 = 8

56³: (5 + 6)³ = 11³ = 1331; 1 + 3 + 3 + 1 = 8

Solution: The answer is 56.

SPEED BUMP

Don’t use this process for anything but perfect cubes, because you won’t get the correct answer. Also, restrict this process to cubes of six digits or less; working larger numbers using this method can be an exhausting process!

The Least You Need to Know

• A perfect square should end in 1, 4, 5, 6, or 9.
• To find the square root of any number, divide by two times the first digit of your answer and subtract by the duplexes.
• If a perfect cube ends in 1, 4, 5, 6, or 9, the root should end with the same number. If the cube ends in 2, the root should end in 8, and vice versa. If the cube ends in 3, the root should end in 7, and vice versa.
• The digit sum can help you confirm that the answer you get matches the number.