CHAPTER
12

Numbers to the Fourth and Fifth Powers

In This Chapter

• How to calculate a number to the fourth power
• How to calculate a number to the fifth power
• The significance of the ratio

The ways to calculate a number to the fourth and fifth power aren’t so different from what you learned from squares and cubes in Chapters 10 and 11. I think you’ll be quite surprised at how quickly you pick up the formulas for fourth and fifth powers. You may even find yourself not even needing to write things down!

In this chapter, I give you the formulas for calculating the values of numbers to the fourth and fifth powers and show you how to apply them.

Calculating a Number to the Fourth Power

Like you did for cubes, to find out what a number is to the fourth power, you first plug in the digits to (a + b). Again, a is the tens place of the number and b is the ones place of the number. Let’s see what the expanded formula for (a + b) is when raised to the fourth power:

Like the cube formula, the formula for the fourth power contains the ratio . You get that by dividing a³b by a⁴, a²b² by a³b, and ab³ by b⁴. If you recall from Chapter 11, you multiply each term in the first line by this ratio to help you find the value of a number to the fourth power.

Now consider the second line. Here, a³b and ab³ have been tripled to 3a³b and 3ab³, and a²b² has been multiplied five times to make it 5a²b². Added together, these two lines give you the expression for (a + b)⁴, which is a⁴ + 4a³b + 6a²b² 4ab³ + b⁴. That means you can find the numbers for the two lines and combine them to find a two-digit number to the fourth power.

The following are some examples that show you how to plug in to the expanded formula to get the value of a number raised to the fourth power.

Example 1

Find the value of 12⁴.

Step 1: Here, a is 1 and b is 2, so is , or 2. Raise a to the fourth power: 1⁴ = 1. This 1 is the first number in the line.

12⁴ = 1

Step 2: Find the rest of the numbers for the first line as set up in the expansion of (a + b)⁴. Multiply 1 and : 1 × 2 = 2. Place the 2 to the right of the 1. Because is , multiply each subsequent digit by 2 until you get to the b⁴ value: 2 × 2 = 4, 4 × 2 = 8, and 8 × 2 = 16. Because 16 is equal to b⁴, your first line is complete.

Step 3: To get the second line, multiply the second term by 3, the third term by 5, and the fourth term by 3, as you saw in the expansion: 2 × 3 = 6, 4 × 5 = 20, and 8 × 3 = 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, put down 6 and carry over the 1 to the next step.

Step 5: In the next column, add 8, 24, and the carryover: 8 + 24 + 1 = 33. Put down 3 and carry over the 3 to the next step.

Step 6: In the next column, add 4, 20, and the carryover: 4 + 20 + 3 = 27. Put down 7 and carry over the 2 to the next step.

Step 7: In the next column, add the 2, 6, and carryover: 2 + 6 + 2 = 10. Put down 0 and carry over the 1 to the next step.

Step 8: In the last column, add 1 and the carryover: 1 + 1 = 2. Put down 2.

Solution: The answer is 20,736.

QUICK TIP

If you aren’t sure what would be considered the b⁴ value and therefore don’t know when to stop multiplying, think of it this way: you only need five values for the first line. So once you get to the fifth number, you can move on to finding the numbers for the second line.

Example 2

Find the value of 13⁴.

Step 1: Here, a is 1 and b is 3. Take a, which is 1, and raise it to the fourth power: 1⁴ = 1. This 1 is the first number in the line.

13⁴ = 1

Step 2: Find the rest of the numbers for the first line as it is in the expansion of (a + b)⁴. Because is , you can get the first line of the problem by multiplying each subsequent number by 3 until you get the b⁴ value: 1 × 3 = 3, 3 × 3 = 9, 9 × 3 = 27, and 27 × 3 = 81. The last number, 81, is equal to b⁴, so the first line is complete.

Step 3: To get the second line, multiply the second term by 3, the third term by 5, and the fourth term by 3, as you saw in the expansion: 3 × 3 = 9, 9 × 5 = 45, and 27 × 3 = 81.

Step 4: Add from right to left, making sure every column before the last only has one digit in it. In the first column, for 81, put down 1 and carry over the 8 to the next step.

Step 5: In the next column, add 27, 81, and the carryover: 27 + 81 + 8 = 116. Put down 6 and carry over the 11 to the next step.

Step 6: In the next column, add 9, 45, and the carryover: 9 + 45 + 11 = 65. Put down 5 and carry over the 6 to the next step.

Step 7: In the next column, add 3, 9, and the carryover: 3 + 9 + 6 = 18. Put down 8 and carry over the 1 to the next step.

Step 8: In the last column, add 1 and the carryover: 1 + 1 = 2. Put down 2.

Solution: The answer is 28,561.

Example 3

Find the value of 32⁴.

Step 1: Here, a is 3 and b is 2. Take a, which is 3, and raise it to the fourth power: 3⁴ = 81. The 81 is the first number in the line.

32⁴ = 81

Step 2: Find the rest of the numbers for the first line as it is in the expansion of (a + b)⁴ until you get to the b⁴ value. Because is , you can get our first line of the problem by multiplying each subsequent number by 2 and dividing by 3: (81 × 2) ÷ 3 = 54, (54 × 2) ÷ 3 = 36, (36 × 2) ÷ 3 = 24, and (24 × 2) ÷ 3 = 16. The last number, 16, is equal to b⁴, so the first line is complete.

Step 3: To get the second line, multiply the second term by 3, the third term by 5, and the fourth term by 3, as you saw in the expansion: 54 × 3 = 162, 36 × 5 = 180, and 24 × 3 = 72.

Step 4: Add right to left, making sure every column before the last only has one digit in it. In the first column, for 16, put down 6 and carry over the 1 to the next step.

Step 5: In the next column, add 24, 72, and the carryover: 24 + 72 + 1 = 97. Put down 7 and carry over the 9 to the next step.

Step 6: In the next column, add 36, 180, and the carryover: 36 + 180 + 9 = 225. Put down 5 and carry over the 22 to the next step.

Step 7: In the next column, add 54, 162, and the carryover: 54 + 162 + 22 = 238. Put down 8 and carry over the 23 to the next step.

Step 8: In the last column, add 81 and the carryover: 81 + 23 = 104. Put down 104.

Solution: The answer is 1,048,576.

Calculating a Number to the Fifth Power

The method for calculating what a number is to the fifth power is very much like what you did for the fourth power. First, let’s see what the expanded formula is for (a + b)⁵:

(a + b)⁵ = a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵

Broken down further, here’s what the first line looks like:

a⁵ + a⁴b + a³b² + a²b³ + ab⁴ + b⁵

Like the formula for the fourth power, the ratio of holds true here as well.

The second line of the formula is this:

4a⁴b + 9a³b² + 9a²b³ + 4ab4

So (a + b)⁵ should look like the following:

And that’s it! Try out this formula by working through the following examples.

Example 1

Find the value of 12⁵.

Step 1: Here, a is 1 and b is 2. Take the a value and raise it to the fifth power: 1⁵ = 1. The 1 is the first number in the line.

12⁵ = 1

Step 2: Find the rest of the numbers for the first line as it is in the expansion of (a + b)⁵ until you get to the b⁵ value. Because is , multiply each subsequent number by 2: 1 × 2 = 2, 2 × 2 = 4, 4 × 2 = 8, 8 × 2 = 16, and 16 × 2 = 32. The last number, 32, is equal to b⁵, so the first line is complete.

Step 3: To get the second line, multiply the second term by 4, the third term by 9, and the fourth term by 9, and the fifth term by 4, as you saw in the expansion: 2 × 4 = 8, 4 × 9 = 36, 8 × 9 = 72, and 16 × 4 = 64.

Step 4: Add right to left, making sure every column before the last only has one digit in it. In the first column, put down 2 and carry over the 3 to the next step.

Step 5: In the next column, add 16, 64, and the carryover: 16 + 64 + 3 = 83. Put down 3 and carry over the 8 to the next step.

Step 6: In the next column, add 8, 72, and the carryover: 8 + 72 + 8 = 88. Put down 8 and carry over the other 8 to the next step.

Step 7: In the next column, add 4, 36, and the carryover: 4 + 36 + 8 = 48. Put down 8 and carry over the 4 to the next step.

Step 8: In the next column, add 2, 8, and the carryover: 2 + 8 + 4 = 14. Put down 4 and carry over the 1 to the next step.

Step 9: In the last column, add 1 and the carryover: 1 + 1 = 2. Put down 2.

Solution: The answer is 248,832.

SPEED BUMP

Don’t forget, if the a digit isn’t 1, you can’t simply multiply by b and have your answer; because you’re using the ratio , you have to multiply by b and divide by a.

The Least You Need to Know

• To find out how much a number is to the fourth power, use a⁴ + a³b + a²b² + ab³ + b⁴ and 3a³b +5a²b² + 3ab³ to find the values, and add right to left.
• To do fifth power calculations, use a⁵ + a⁴b + a³b² + a²b³ + ab⁴ + b⁵ and 4a⁴b + 9a³b² + 9a²b³ + 4ab⁴ to find the values, and add right to left.
• Use to find the numbers for the first line of the formula.