Math CentralQuandaries & Queries


Subject: Cube rootz
Name: Jacqui

Who are you: Student

How would you find the answer of a cube root backwards. For example: 8000 but backwards. Like, what number made that and how would you do it? I'm really confused.
could you please help me?

Hi Jacqui,

This is a useful technique to learn because it allows you to estimate cube roots or other roots. The number you have is an 8 followed by three zeros and I know that 10 x 10 x 10 = 103 = 1000 so I think that the cube root of 8000 is probably a two digit number with the units digit 0, that is a number d0 where d is a one digit number.

So what is d to give me a cube of 8000? I know that 2 x 2 x 2 = 8 so 203 = 8000.

But 8000 is quite special, it is the cube if an integer whereas most integers are not cubes of integers. For example what is the cube root of 9413? I don't know but I can find an estimate quite easily. As I observed with 8000 the cube root of 9413 is approximately a two digit number. (9413 = 9000 + 413 and the 3 zeros tell me to try d0 again.) 203 = 8000 and 303 = 27000 so the cube root of 9413 is between 20 and 30 and probably much close to 20 than 30. So far I have only used mental arithmetic. Next I tried 223 and calculated it with paper and pencil and found that 223 = 10648. Hence the cube root of 9413 is largest than 20 and smaller than 22. If you need a better estimate try 213 and see if the cube root of 9413 is larger than or smaller than 21.


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