helpful for solubility

what's a quick way to determine something like

(5 X 10^-7)^1/3

?

ex (4x^3 = 3.1 X 10^-21) solve for x

To find the cube root of something, you want the exponent to be divisible by 3. The -7 should either be changed to -6 by decreasing 5 to 0.5 or changed to -9 by increasing 5 to 500. I don't know the cube root of what 0.5 is but I can probably approximate the cube root of 500. 5x5x5 =125 and 10x10x10=1000 so 500 must be somewhere between 5 and 10. I'll go ahead and say 7. Therefore the cube root of 500 is 7. The cube root of 10^-9 is just 10^-3. The final answer would be 7x10^-3 M or very close to that.

You can use similar logic with 4x^3 = 3x10^-31. Isolating x^3 yields: 0.75 x 10^-31. I don't know the cube root of any decimal but I can approximate a whole number. I want to make 0.75 greater than 1 but at the same time, make sure the exponent is divisible by 3:

0.75x10^-31 = 75x10^-33

The cube root of 10^-33 is 10^-11. If 3x3x3=27 and 5x5x5=125. The cube root of 75 must be about 4 or very close to that. The final answer should be around 4x10^-11.