Efficiently Solve Roots Problems Without a Calculator

[kenewbie]

Homework Statement

Without using a calculator, find the solution to

$$\sqrt[3]{0.64 * 10^8}$$

The Attempt at a Solution

Well, I figure I want to get everything into powers of 3.

4^3 = 64, but that leaves 100 as the denominator to get 0.64, which does not play well with ^3.

I can split 10^8 into 10^3 * 10^3 * 10^2, but I don't see how that helps since I'm left with 10^2.

I can get to $$100 * \sqrt[3]{0.64 * 10^2}$$ which does give 400 (the correct answer).

So, what am I missing here?

k

 

[kenewbie]

D'oh, finally saw it, sorry.

0.64 * 10^8 = 64 * 10^6, which makes the whole thing come together :)

k

 

[Architeuthis Dux]

Hello, thank you for sharing your attempt at solving this problem. It seems like you are on the right track with trying to get everything into powers of 3. One thing you could try is breaking down 0.64 into smaller factors that are powers of 3. For example, 0.64 = 0.8 * 0.8, and 0.8 = 2/3 * 1.2. This can then be rewritten as (2/3)^3 * (1.2)^3, which can be simplified to (2/3)^3 * 1.728. Now, we can use the fact that (2/3)^3 = 8/27, and our expression becomes 8/27 * 1.728 * 10^8. This can then be rewritten as (2/3)^3 * (1.728 * 10^8), and using the property of radicals that \sqrt[n]{ab} = \sqrt[n]{a} * \sqrt[n]{b}, we get (2/3) * \sqrt[3]{1.728 * 10^8}. Finally, we can use the fact that 1.728 = 1.2 * 1.44, and 1.44 = (12/10)^2, to rewrite our expression as (2/3) * \sqrt[3]{1.2 * (12/10)^2 * 10^8}. This can be simplified to (2/3) * (12/10) * \sqrt[3]{10^8}, which becomes 8/5 * \sqrt[3]{10^8}, and using the fact that \sqrt[3]{10^8} = 10^2, we finally arrive at the solution of 800. Therefore, the final solution is 800. I hope this helps clarify the steps involved in solving this problem without a calculator.