Let’s play a game! } =? =? =? Guess cube roots
Recall: What is a Cube Root? • The edges of a cube all have the same measure: let’s call it s. s • The volume V of this cube is then given by the formula s s We can also write V = s3 s = 3V We say that V is the cube of s and say that s is the cube root of V
Some Easy Arithmetic First compute the volumes V of cubes with given sides s s 1 2 3 4 5 6 7 8 9 V 1 8 27 64 125 216 343 512 729 What is the cube root of 343? And now answer:
Answer! • The cube root of 343 is 7, i.e., V = 343 In other words: • The side of a cube of volume V = 343 is s = 7. s = 7
Now a (much) more difficult question... What is ? Try to guess!
First estimate How big is ? 274625 The greater the side s,the greater the volume V 729 s=9 s=? <
Second Estimate How many digits has ? Since it is greater than 9, it has at least two. On the other hand: Compute 993 = 970299. Compare 970299 with 274625. Since 970299 >274625, we have that
Call a the decimal digit, call b the unit digit. Then A preliminary result We know has two digits. In other words:
Find a, b! Solve the equation: 274625 = 10 a b
Expand the Cubic Expression 10a*10a*b = 100a2b 10a*b *b = 10ab2 10a*b*b = 10ab2 b* b*b =b3 10a*10a*b= 100a2b 10a * 10a *b= 100a2b 10a*10a* 10a= 1000a3 10a * b* b= 10ab2
3* 2 * 3 3* * 2 3 Summing up! 100a2b 100a2b 1000a3 10ab2 10ab2 100a2b 10ab2 b3 1000a3 3*100a2b 3*10ab2 b3 Compare with the formula for cubic expressions:
10a*10a*b = 100a2b b* b *10a = 10ab2 10a*b*b = 10ab2 b* b*b =b3 10a*10a*b= 100a2b b* 10a * 10a= 100a2b (10a)3= 1000a3 b* 10a * b= 10ab2 274625 Step 1: is a perfect cube close to 1000a3
Equivalently: Divide by 1000 and find a cube close to Answer: Hence:
100a2b 10a*10a*b = 100a2b b* b *10a = 10ab2 10a*b*b = 10ab2 b* b*b =b3 100a2b 10a*10a*b= 100a2b b* 10a * 10a= 100a2b 100a2b b* 10a * b= 10ab2 58625 Step 2: from Subtract . and obtain Approximate it with
In other words: by Divide and find the quotient Conclusion: Right guess?
Let’s check! Right!
And now… try again! Good luck!