Question

Rationalise the denominator of \[\dfrac{{\sqrt {40} }}{{\sqrt 3 }}\]

Answer 1

Hint: For rationalisation we have to multiply the fraction in such a way that the resulting fraction is always an equivalent fraction of the same. Multiply \[\dfrac{{\sqrt {40} }}{{\sqrt 3 }}\] with such a fraction that there won't be a root in the denominator.
Completed Step by Step Solution:
Keeping in mind that the resulting fraction after rationalisation must be an equivalent fraction of what we are trying to rationalise. We can always multiply the irrational term in the denominator in both numerator and denominator which in this case is \[\sqrt 3 \] so if we multiply the given fraction with \[\dfrac{{\sqrt 3 }}{{\sqrt 3 }}\]
Which is basically 1 and we know if we multiply any number with 1 it won’t hamper the result. Thus it will be written as
\[\begin{array}{l}
 = \dfrac{{\sqrt {40} }}{{\sqrt 3 }} \times \dfrac{{\sqrt 3 }}{{\sqrt 3 }}\\
 = \dfrac{{\sqrt {40} \times \sqrt 3 }}{{\sqrt 3 \times \sqrt 3 }}\\
 = \dfrac{{\sqrt {40 \times 3} }}{{\sqrt {3 \times 3} }}\\
 = \dfrac{{\sqrt {4 \times 10 \times 3} }}{{\sqrt {{3^2}} }}\\
 = \dfrac{{\sqrt {{2^2} \times 30} }}{{\sqrt {{3^2}} }}
\end{array}\]
Now we can see that \[\sqrt {{2^2}} = 2\& \sqrt {{3^2}} = 3\]
Therefore, We get
\[ = \dfrac{{2\sqrt {30} }}{3}\]

Note: One must be conscious about the equivalent fraction or the answer will change; we must multiply with the same number in both numerator and denominator. We must also note, why rationalisation is necessary because we cannot divide a number by an irrational number so in any question if we leave a solution without rationalising it to the end it will still be counted as an incomplete solution.