Between what two consecutive integers do root3(150) lie?

Oct 15, 2015

5 and 6

Explanation:

$\sqrt[3]{150} = 5.31$ to 2 dp

$5 < 5.31 < 6$

Oct 15, 2015

$5 < \sqrt[3]{150} < 6$

Explanation:

${5}^{3} = 125$
and
${6}^{3} = 216$

${5}^{3} < 150 < {6}^{3}$

$\Rightarrow 5 < \sqrt[3]{150} < 6$

This range can be found by experimentation:
- pick two numbers one too large and one too small (eg. 0 and 10, in this case);
- take their average (5);
- compute it's cube;
- replace the smaller number if the cube is less than 150 or the larger if the cube is greater than 150
repeat until the difference between the small and large is $\le 1$.

OR
use a calculator to determine
$\sqrt[3]{150} = {150}^{\frac{1}{3}} \cong 5.313$ (I had to use the fractional exponent version for Excel)
$\Rightarrow 5 < 5.313 < 6$