Extracting Fifth Roots MentallyPresented by Grey Matters Mental Gym

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Extracting Fifth Roots Mentally

• A fifth power is a number multiplied by itself four more times. For example, 3 to the fifth power (35) is 3 * 3 * 3 * 3 * 3, or 243.

• A fifth root is the number that, when taken to the fifth power, results in a given number. For example, the fifth root of 243 (5√243) is 3.

Extracting Fifth Roots Mentally

• Have a spectator choose any number from 1 to 100, taking it to the fifth power with their calculator and give you the answer.

Imagine being able to figure the fifth root of that number in your head!

Extracting Fifth Roots Mentally Requirements

• A calculator capable of displaying 11 or more digits

Spectator

Extracting Fifth Roots Mentally

• To be able to extract fifth roots mentally, you must know the first 10 mentally:

• 15=1 25=32• 35=243 45=1,024• 55=3,125 65=7,776• 75=16,807 85=32,768

95=59,049 105=100,000

Extracting Fifth Roots Mentally

• Note that each fifth power ends in the same digit as its fifth root:

• 15=1 25=32• 35=243 45=1,024• 55=3,125 65=7,776• 75=16,807 85=32,768• 95=59,049 105=100,000

Extracting Fifth Roots Mentally

If you’re given one of the fifth power you’ve memorized from this chart, simply give the fifth root you’ve remembered.

Extracting Fifth Roots Mentally

• If you’re given a number that ranges from 100,000 (105) to 10,000,000,000 (1005), you’ll need to go through the following steps.

Extracting Fifth Roots Mentally

First, split the number into two parts, with the right half consisting of the 5 rightmost digits, and the left half consisting of the rest. As an example, let’s say you’re given the number 11,881,376. You would mentally split it into “118” and “81,376”.

Extracting Fifth Roots Mentally

• Next, focusing on the number to the left of the comma, ask yourself, “What is the largest fifth power that is equal to or less than that number?”

• In our example, the largest fifth power that is equal to or less than 118 is 32.

Extracting Fifth Roots Mentally

• Recall the fifth root of this number, and that will be the tens digit of the answer.

• In our example, we found that 32 was the largest fifth power less than or equal to 118. Since we know that the fifth root of 32 is 2 (5√32=2), we now know our answer is in the 20s somewhere.

Extracting Fifth Roots Mentally

Now, focus on the other number. In our example of 11,881,376, this would be the “81,376”.

Extracting Fifth Roots Mentally

• Look at just the rightmost digit. This will be the digit in the ones place of the answer!

In our example of 11,881,376, we see that the rightmost digit is a 6. Therefore, the ones digit of the answer is also a 6!

Extracting Fifth Roots Mentally

• Once you know both the tens digit and the ones digit of the answer, put them together and you have the fifth root.

In our example, we determined that the answer for “11,881,376” is in the 20s, and that it ends in 6, so we give 26 as the fifth root!

Extracting Fifth Roots Mentally

• Other examples:• What is 5√147,008,443? Because “1470”

is greater than 1,024 (45), and less than 3,125 (55), we know the answer is in the 40s. Because “08443” ends in 3, we know that the answer also ends in 3. Therefore, 5√147,008,443=43.

Extracting Fifth Roots Mentally

• Other examples:• What is 5√1,934,917,632? Because

“19349” is greater than 16,807 (75), and less than 32,768 (85), we know the answer is in the 70s. Because “17632” ends in 2, we know that the answer also ends in 2. Therefore, 5√1,934,917,632=72.

Extracting Fifth Roots Mentally

• Other examples:• What is 5√8,587,340,257? Because

“85,873” is greater than 59,049 (95), and less than 100,000 (105), we know the answer is in the 90s. Because “40257” ends in 7, we know that the answer also ends in 7. Therefore, 5√8,587,340,257=97.