Question

\[\sqrt 3 = 1.732\], \[\sqrt 5 = 2.236\] and \[\sqrt {10} = 3.162\] find
i) \[\dfrac{1}{{\sqrt[4]{3} - \sqrt[3]{5}}}\] ii) \[\dfrac{{3 + \sqrt 5 }}{{3 - \sqrt 5 }}\]

Answer 1

Hint: The square root is a number that is exactly double to it, it means let us take a number 4 and the square root of the number 4 is written as \[\sqrt 4 \], then the value of it is 2 that is exactly half of 4. Same way there will be a cube root which can be written as \[\sqrt[3]{4}\]. In the given question, the value of the square root of 3, 5 and 10 is given, just substitute the values in the given question and solve to get the result.

Complete step-by-step answer:
The given values of square root is \[\sqrt 3 = 1.732\], \[\sqrt 5 = 2.236\] and \[\sqrt {10} = 3.162\].
i) \[\dfrac{1}{{\sqrt[4]{3} - \sqrt[3]{5}}}\]
In order to find the value of equation \[\dfrac{1}{{\sqrt[4]{3} - \sqrt[3]{5}}}\], we will simplify the equation.
\[\begin{array}{c}
\dfrac{1}{{\sqrt[4]{3} - \sqrt[3]{5}}} = \dfrac{1}{{1.31 - 1.71}}\\
 = \dfrac{1}{{ - 0.4}}\\
 = - 2.5
\end{array}\]
Therefore, the value of \[\dfrac{1}{{\sqrt[4]{3} - \sqrt[3]{5}}}\] is -2.5.
ii) \[\dfrac{{3 + \sqrt 5 }}{{3 - \sqrt 5 }}\]
In order to find the value of equation \[\dfrac{1}{{\sqrt[4]{3} - \sqrt[3]{5}}}\], we will simplify the equation by putting he value \[\sqrt 5 = 2.236\].
\[\begin{array}{c}
\dfrac{{3 + \sqrt 5 }}{{3 - \sqrt 5 }} = \dfrac{{3 + 2.236}}{{3 - 2.236}}\\
 = \dfrac{{5.236}}{{0.764}}\\
 = 6.853
\end{array}\]
Therefore, the value of \[\dfrac{{3 + \sqrt 5 }}{{3 - \sqrt 5 }}\]is \[6.853\].

Note: Here, finding the value of the \[\sqrt[4]{3} - \sqrt[3]{5}\]will quite difficult, so be careful while solving it. The equation \[\sqrt[4]{3} - \sqrt[3]{5}\] can be written as \[{3^{\dfrac{1}{4}}} - {5^{\dfrac{1}{3}}}\] and we can find the value of the each term, then we can subtract the both terms to get the value of the given question.