Question

The product of $\sqrt 3 {\text{ and }}\sqrt[3]{5}$ is?
$\left( a \right)\sqrt[6]{{375}}$
$\left( b \right)\sqrt[6]{{675}}$
$\left( c \right)\sqrt[6]{{575}}$
$\left( d \right)\sqrt[6]{{475}}$

Answer 1

Hint: In this particular question use the concept of the root conversion by taking the L.C.M of the order of the roots and change the order of the given roots according to the LCM value, so that order of each root becomes sane so use these concepts to reach the solution of the question.

Complete step-by-step answer: Given roots
$\sqrt 3 {\text{ and }}\sqrt[3]{5}$
Now we have to multiply these roots.
$ \Rightarrow \sqrt 3 \times \sqrt[3]{5} = $ ?
So first root is,
$\sqrt 3 $
Now as we know this is a square root so the order of the root is 2.
Other given root
$\sqrt[3]{5}$
Now as we know this is a cube root so the order of the root is 3.
Now take L.C.M of the orders of the given roots.
So the L.C.M of 2 and 3 is ($2 \times 3$) = 6.
So convert the order of the given roots into 6 so that we can easily multiply each other.
Therefore, $\sqrt 3 = {\left( 3 \right)^{\dfrac{1}{2}}}$
Now multiply and divide by 3 in the power of the above expression so that the order of the square root changes to 6.
$ \Rightarrow \sqrt 3 = {\left( 3 \right)^{\dfrac{1}{2}}} = {\left( 3 \right)^{\dfrac{3}{6}}} = \sqrt[6]{{{3^3}}} = \sqrt[6]{{27}}$................. (1)
Now change the order of the other root by same method we have,
Therefore, $\sqrt[3]{5} = {\left( 5 \right)^{\dfrac{1}{3}}}$
Now multiply and divide by 2 in the power of the above expression so that the order of the cube root changes to 6.
$ \Rightarrow \sqrt[3]{5} = {\left( 5 \right)^{\dfrac{1}{3}}} = {\left( 5 \right)^{\dfrac{2}{6}}} = \sqrt[6]{{{5^2}}} = \sqrt[6]{{25}}$.......................... (2)
Now multiply equations (1) and (2) we have,
$ \Rightarrow \sqrt 3 \times \sqrt[3]{5} = \sqrt[6]{{27}} \times \sqrt[6]{{25}} = \sqrt[6]{{27 \times 25}} = \sqrt[6]{{675}}$
So this is the required answer.

So, the correct answer is “Option B”.


Note: Whenever we face such types of questions the key concept we have to remember is that always recall that that if the order of two roots are not equal then we cannot multiply them directly so that it become a single root, so we have to change the order of the given roots so that it become equal as above and then multiply we will get the required answer.