Question

Describe how to estimate a non-perfect square root to the hundredths place without using a calculator.

Answer 1

Given :

A non perfect square root number.

To find :

Square root to hundredths place.

Solution :

A number is given which is not a perfect square root number, it means the numbers are not,

1² ,2² ,3² , 4² ... n² where n is an integer.

so, lets make a function of square root.

y = f(x)

$\bf \: y = f(x) = {x}^{ \frac{1}{2} }$

(equation 1)

So,

For finding the square root of a non square number, we take the differentiation of function as,

$\\ \bf \: \frac{dy}{dx} = \frac{1}{2( {x}^{ \frac{1}{2} }) }$

(equation 2 )

so,

$\bf \: y \: = y _{0} + ( \Delta x) \frac{dy}{dx}$

where y0 = square root of nearest number which is a perfect square root.

and ∆x = Difference between non perfect square root number and perfect square root number.

Putting value of equation 2 in 3,

For example :

Lets find the square root of 40,

So nearest number which has perfect square root = 36 whose square root = 6

So

y0 = 6

so,

$\bf \: \sqrt{40} = \sqrt{36} + ( \frac{40 - 36}{2 \times \sqrt{36} } )$

$\bf \sqrt{40} = 6 + \frac{4}{12} \\ \\ \: \: \: \: \: \: \: \: \: \: \bf \: = 6.33$