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# Extracting Cube Roots Mentally

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

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### Extracting Cube Roots Mentally

1. 1. Extracting Cube Roots Mentally <ul><li>Presented by Grey Matters Mental Gym </li></ul><ul><li>http://members.cox.net/beagenius/ </li></ul>
2. 2. Extracting Cube Roots Mentally <ul><li>A cube is a number multiplied by itself twice more. For example, 3 cubed (3 3 ) is 3 * 3 * 3, or 27. </li></ul><ul><li>A cube root is the number that, when cubed, results in a given number. For example, the cube root of 27 ( 3 √27) is 3. </li></ul>
3. 3. Extracting Cube Roots Mentally <ul><li>Have a spectator choose any number from 1 to 100, cube it with their calculator and give you the answer. </li></ul><ul><li>Imagine being able to figure the cube root of that number in your head! </li></ul>
4. 4. Extracting Cube Roots Mentally Requirements <ul><li>Calculator </li></ul><ul><li>Spectator </li></ul>
5. 5. Extracting Cube Roots Mentally <ul><li>To be able to extract cube roots mentally, you must know the first 10 cubes mentally: </li></ul><ul><li>1 3 =1 2 3 =8 </li></ul><ul><li>3 3 =27 4 3 =64 </li></ul><ul><li>5 3 =125 6 3 =216 </li></ul><ul><li>7 3 =343 8 3 =512 </li></ul><ul><li>9 3 =729 10 3 =1000 </li></ul>
6. 6. Extracting Cube Roots Mentally <ul><li>Note that no two cubes end in the same digit: </li></ul><ul><li>1 3 = 1 2 3 = 8 </li></ul><ul><li>3 3 =2 7 4 3 =6 4 </li></ul><ul><li>5 3 =12 5 6 3 =21 6 </li></ul><ul><li>7 3 =34 3 8 3 =51 2 </li></ul><ul><li>9 3 =72 9 10 3 =100 0 </li></ul>
7. 7. Extracting Cube Roots Mentally <ul><li>Also notice that the cubes of 1, 4, 5, 6, 9 and 0 end in those same digits: </li></ul><ul><li>1 3 = 1 2 3 =8 </li></ul><ul><li>3 3 =27 4 3 =6 4 </li></ul><ul><li>5 3 =12 5 6 3 =21 6 </li></ul><ul><li>7 3 =343 8 3 =512 </li></ul><ul><li>9 3 =72 9 10 3 =100 0 </li></ul>
8. 8. Extracting Cube Roots Mentally <ul><li>Finally, notice that 2 3 ends in 8, 8 3 ends in 2, 3 3 ends in 7 and 7 3 ends in 3: </li></ul><ul><li>1 3 =1 2 3 = 8 </li></ul><ul><li>3 3 =2 7 4 3 =64 </li></ul><ul><li>5 3 =125 6 3 =216 </li></ul><ul><li>7 3 =34 3 8 3 =51 2 </li></ul><ul><li>9 3 =729 10 3 =1000 </li></ul>
9. 9. Extracting Cube Roots Mentally <ul><li>If you’re given one of the cubes you’ve memorized from this chart, simply give the cube root you’ve remembered. </li></ul>
10. 10. Extracting Cube Roots Mentally <ul><li>If you’re given a number that ranges from 1,000 (10 3 ) to 1,000,000 (100 3 ), you’ll need to go through the following steps. </li></ul>
11. 11. Extracting Cube Roots Mentally <ul><li>First, split the number into two parts at the comma. As an example, let’s say you’re given the number 39,304. You would mentally split it into “39” and “304”. </li></ul>
12. 12. Extracting Cube Roots Mentally <ul><li>Next, focusing on the number to the left of the comma, ask yourself, “What is the largest cube that is equal to or less than that number?” </li></ul><ul><li>In our example, the largest cube that is equal to or less than 39 (the number to the left of the comma) is 27. </li></ul>
13. 13. Extracting Cube Roots Mentally <ul><li>Recall the cube root of this number, and that will be the tens digit of the answer. </li></ul><ul><li>In our example, we found that 27 was the largest cube less than or equal to 39. Since we know that the cube root of 27 is 3 ( 3 √27=3), we now know our answer is in the 30s somewhere. </li></ul>
14. 14. Extracting Cube Roots Mentally <ul><li>Now, focus on the number to the right of the comma. In our example of 39,304, this would be the “304”. </li></ul>
15. 15. Extracting Cube Roots Mentally <ul><li>Look at just the rightmost digit, and ask yourself, “Which digit, from 0-9, when cubed, would result in that digit?” This will be the ones digit of the answer. </li></ul><ul><li>In our example of 39,304, we see that the rightmost digit is a 4. Remembering that 4 is the only cube that ends in a 4 (4 3 =64), this must be the ones digit of the answer! </li></ul>
16. 16. Extracting Cube 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 cube root. </li></ul><ul><li>In our example, we determined that the answer for “39,304” is in the 30s, and that it ends in 4, so we give 34 as the cube root! </li></ul>
17. 17. Extracting Cube Roots Mentally <ul><li>Other examples: </li></ul><ul><li>What is 3 √50,653? Because “50” is greater than 27 (3 3 ), and less than 64 (4 3 ), we know the answer is in the 30s. Because “653” ends in 3, we know that the answer must end in 7, as only 7 3 (343) ends in 3. Therefore, 3 √50,653=37. </li></ul>
18. 18. Extracting Cube Roots Mentally <ul><li>Other examples: </li></ul><ul><li>What is 3 √314,432? Because “314” is greater than 216 (6 3 ), and less than 343 (7 3 ), we know the answer is in the 60s. Because “432” ends in 2, we know that the answer must end in 8, as only 8 3 (512) ends in 2. Therefore, 3 √314,432=68. </li></ul>
19. 19. Extracting Cube Roots Mentally <ul><li>Other examples: </li></ul><ul><li>What is 3 √704969? Because “704” is greater than 512 (8 3 ), and less than 729(9 3 ), we know the answer is in the 80s. Because “969” ends in 9, we know that the answer must end in 9, as only 9 3 (729) ends in 9. Therefore, 3 √704969=89. </li></ul>