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# Extracting Fifth Roots Mentally

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How do you find fifth roots, using only your mind? This slideshow will show you how!

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### Extracting Fifth Roots Mentally

1. 1. Extracting Fifth Roots Mentally <ul><li>Presented by Grey Matters Mental Gym </li></ul><ul><li>http://members.cox.net/beagenius/ </li></ul>
2. 2. Extracting Fifth Roots Mentally <ul><li>A fifth power is a number multiplied by itself four more times. For example, 3 to the fifth power (3 5 ) is 3 * 3 * 3 * 3 * 3, or 243. </li></ul><ul><li>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. </li></ul>
3. 3. Extracting Fifth Roots Mentally <ul><li>Have a spectator choose any number from 1 to 100, taking it to the fifth power with their calculator and give you the answer. </li></ul><ul><li>Imagine being able to figure the fifth root of that number in your head! </li></ul>
4. 4. Extracting Fifth Roots Mentally Requirements <ul><li>A calculator capable of displaying 11 or more digits </li></ul><ul><li>Spectator </li></ul>
5. 5. Extracting Fifth Roots Mentally <ul><li>To be able to extract fifth roots mentally, you must know the first 10 mentally: </li></ul><ul><li>1 5 =1 2 5 =32 </li></ul><ul><li>3 5 =243 4 5 =1,024 </li></ul><ul><li>5 5 =3,125 6 5 =7,776 </li></ul><ul><li>7 5 =16,807 8 5 =32,768 </li></ul><ul><li>9 5 =59,049 10 5 =100,000 </li></ul>
6. 6. Extracting Fifth Roots Mentally <ul><li>Note that each fifth power ends in the same digit as its fifth root: </li></ul><ul><li>1 5 = 1 2 5 =3 2 </li></ul><ul><li>3 5 =24 3 4 5 =1,02 4 </li></ul><ul><li>5 5 =3,12 5 6 5 =7,77 6 </li></ul><ul><li>7 5 =16,80 7 8 5 =32,76 8 </li></ul><ul><li>9 5 =59,04 9 10 5 =100,00 0 </li></ul>
7. 7. Extracting Fifth Roots Mentally <ul><li>If you’re given one of the fifth power you’ve memorized from this chart, simply give the fifth root you’ve remembered. </li></ul>
8. 8. Extracting Fifth Roots Mentally <ul><li>If you’re given a number that ranges from 100,000 (10 5 ) to 10,000,000,000 (100 5 ), you’ll need to go through the following steps. </li></ul>
9. 9. Extracting Fifth Roots Mentally <ul><li>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”. </li></ul>
10. 10. Extracting Fifth Roots Mentally <ul><li>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?” </li></ul><ul><li>In our example, the largest fifth power that is equal to or less than 118 is 32. </li></ul>
11. 11. Extracting Fifth Roots Mentally <ul><li>Recall the fifth root of this number, and that will be the tens digit of the answer. </li></ul><ul><li>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. </li></ul>
12. 12. Extracting Fifth Roots Mentally <ul><li>Now, focus on the other number. In our example of 11,881,376, this would be the “81,376”. </li></ul>
13. 13. Extracting Fifth Roots Mentally <ul><li>Look at just the rightmost digit. This will be the digit in the ones place of the answer! </li></ul><ul><li>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! </li></ul>
14. 14. Extracting Fifth Roots Mentally <ul><li>Once you know both the tens digit and the ones digit of the answer, put them together and you have the fifth root. </li></ul><ul><li>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! </li></ul>
15. 15. Extracting Fifth Roots Mentally <ul><li>Other examples: </li></ul><ul><li>What is 5 √147,008,443? Because “1470” is greater than 1,024 (4 5 ), and less than 3,125 (5 5 ), 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. </li></ul>
16. 16. Extracting Fifth Roots Mentally <ul><li>Other examples: </li></ul><ul><li>What is 5 √1,934,917,632? Because “19349” is greater than 16,807 (7 5 ), and less than 32,768 (8 5 ), 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. </li></ul>
17. 17. Extracting Fifth Roots Mentally <ul><li>Other examples: </li></ul><ul><li>What is 5 √8,587,340,257? Because “85,873” is greater than 59,049 (9 5 ), and less than 100,000 (10 5 ), 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. </li></ul>