Question

\[5\sqrt[3]{3}\] represents the pure surd.
(A). \[\sqrt[3]{15}\]
(B). \[\sqrt[3]{375}\]
(C). \[\sqrt[3]{75}\]
(D). \[\sqrt[3]{45}\]

Answer 1

Hint: A surd is the square root (cubic root etc.) of whole numbers which can’t be simplified into a whole number or rational number alone and when the whole rational number is under the radical sign, it’s called a pure surd. A pure surd has no rational factor except 1.

Complete step-by-step answer:

Here in the above question we have been asked to represent \[5\sqrt[3]{3}\] into pure surd.
Now as we know that \[a\times \sqrt[b]{c}\] is equal to \[\sqrt[b]{c\times {{a}^{b}}}\].
We have a = 5, b = 3 and c = 3.
\[\Rightarrow 5\sqrt[3]{3}=\sqrt[3]{3\times {{5}^{3}}}\]
We know that \[{{a}^{n}}=a\times a\times a.....n\] times.
\[\Rightarrow {{5}^{3}}=5\times 5\times 5\]
\[\Rightarrow 5\sqrt[3]{3}=\sqrt[3]{3\times 5\times 5\times 5}=\sqrt[3]{375}\]
Therefore, for the given question the correct answer is option B.

NOTE: Be careful while solving the surd as you can make a silly mistake and get an incorrect answer. Also remember the fact about the surds i.e, we can find a pure surd by adding two surds but the surds to be added must be the same. Also a mixed surd can be converted into pure surd and vice versa. It is one of the facts about the origin of the word ‘surd’ that this word originated from the Latin word ‘surdus’ which means “mute”. This muted sound is largely thought that it represents an irrational number would be a pure, clear sound. Every surd is an irrational number, but every irrational number needs not to be a surd. You may also go through the other properties of a surd which will help you in the complex problem of surds.