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 s s The volume V of this cube is then given by the formula V = s 3 We say that V is the cube of s We can also write s = 3  V 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 V And now answer: What is the cube root of 343?

Answer! The cube root of 343 is 7, i.e., V = 343 s = 7 In other words: The side of a cube of volume V = 343 is s = 7.

Now a (much) more difficult question... What is Try to guess! ?

First estimate How big is ? 729 < The greater the side s, the greater the volume V s=9 s=?

Second Estimate Since it is greater than 9, it has at least two. On the other hand: Compute 99 3 = Compare with Since >274625, we have that How many digits has ?

A preliminary result Call a the decimal digit, call b the unit digit. Then We know In other words: has two digits.

Find a, b! a b Solve the equation: =

10a*10a*b= 100a 2 b Expand the Cubic Expression 10a*10a* 10a= 1000a 3 10a * 10a *b= 100a 2 b 10a*10a*b = 100a 2 b 10a*b*b = 10ab 2 10a * b* b= 10ab 2 10a*b *b = 10ab 2 b*b*b=b3 b*b*b=b3

Summing up! 1000a 3 100a 2 b 10ab 2 b a 3 3*100a 2 b3*10ab 2 b3b3 Compare with the formula for cubic expressions: 3* 2 * 3 3* * 2 3

Step 1: is a perfect cube close to 1000a 3 10a*10a*b= 100a 2 b 10a*10a*b = 100a 2 b (10a) 3 = 1000a 3 10a*b*b = 10ab 2 b* 10a * 10a= 100a 2 b b* b *10a = 10ab 2 b* 10a * b= 10ab 2 b* b*b =b

Equivalently: Divide by 1000 and find a cube close to Answer: Hence:

Step 2: Subtract from 100a 2 b Approximate it with and obtain 10a*10a*b= 100a 2 b 10a*10a*b = 100a 2 b 10a*b*b = 10ab 2 b* 10a * 10a= 100a 2 b b* b *10a = 10ab 2 b* 10a * b= 10ab 2 b* b*b =b

In other words: Divide by Conclusion: and find the quotient Right guess?

Let’s check! Right!

And now… Good luck! try again!