Question

explain ,how can we take cube root of 998 without a calculator

Answer 1

10 cube is 1000. 998 is just a little short of 1000.
So the cube root will be pretty close to 10 such as 9.99

Answer 2

998 has only two prime factors: 2 x 499
so not much help there.

Answer 3

so we will make factor tree ok
now how to find the under root of 917

Answer 4

is there any connection of factors for finding the cube root? as you said 2*499

Answer 5

Sometimes when you write the prime factors you may see a factor repeat itself 3 or more times in which case we can take that factor out of the cube root.

Answer 6

i would do a factor tree and whatever number appears 3 times put it out the radical and what number does appear 3 times keep it inside the radical and then multiple the numbers inside the radical with each other and the number outside with each other and thats it

Answer 7

when number is close to a perfect cube, use a linear approximation
\[f(x) \approx f'(a)(x-a) + f(a)\]
\[\sqrt[3]{998} \approx \frac{1}{3\sqrt[3]{1000}^{2}}(998 - 1000) + \sqrt[3]{1000}\]
\[= -\frac{1}{150} + 10\]

Answer 8

for the linear approximation, you find the slope at the close point f(1000). now you move on the slope instead of the real function to measure change. if you do this you never have to evaluate the cube root of 998 - you just evaluate cube root of 1000, and then subtract the difference of the slope over delta x (the slope has a similar trend as the real function in close region).

calculating the derivative is easier than evaluating cube root of 998. if this is true, then putting the "tangent line" at x=1000 and going along the this tangent line is easier than going along the original function. once the derivative was found, the tangent line is a linear function and can be handled easily.(just multiply change with the slope of the tangent line)