Question

Write ${\left( {625} \right)^{\dfrac{3}{4}}}$ in radical form.

Answer 1

Hint:A radical expression is an expression which contains the radical symbol. Also expressing in radical form simply means simplifying all the radicals present in a radical expression such that in the final equation we have no square root, cube root, 4th roots etc.Also we know: ${\left( {{x^a}} \right)^b} = {x^{ab}}$.So by using the basic definition and the above exponent property we can thus solve the given question.

Complete step by step answer:
Given, ${\left( {625} \right)^{\dfrac{3}{4}}}............................\left( i \right)$. Now our aim is to convert or express the given number in radical form. Such that we have to write ${\left( {625} \right)^{\dfrac{3}{4}}}$ in radical form. So to convert ${\left( {625} \right)^{\dfrac{3}{4}}}$in radical form, let’s first simplify it and express it by using parts. Such that we can write:
${\left( {625} \right)^{\dfrac{3}{4}}} = {\left( {600 + 25} \right)^{\dfrac{3}{4}}}..................\left( {ii} \right)$
Now let’s express $600$in such a way that we would be able to solve this question. So let
 \[600 = \left( {60 \times 10} \right) \\
\Rightarrow 600 = \left( {120 \times 5} \right) \\
\Rightarrow 600 = \left( {12 \times 50} \right) \\
\Rightarrow 600 = \left( {24 \times 25} \right)....................\left( {iii} \right) \\ \]
Now we have expressed $600$ as \[\left( {24 \times 25} \right)\] since then we can take the term $25$ as common and thus further simplify it.
Now substituting (iii) back in (ii) we can write:
\[{\left( {625} \right)^{\dfrac{3}{4}}} = {\left( {600 + 25} \right)^{\dfrac{3}{4}}} \\
\Rightarrow{\left( {625} \right)^{\dfrac{3}{4}}}= {\left( {24 \times 25 + 25} \right)^{\dfrac{3}{4}}}.......................\left( {iv} \right) \\ \]
Now let’s take the term $25$ as a common term and move further.
Such that:
\[{\left( {625} \right)^{\dfrac{3}{4}}} = {\left( {24 \times 25 + 25} \right)^{\dfrac{3}{4}}} \\
\Rightarrow{\left( {625} \right)^{\dfrac{3}{4}}} = {\left( {25\left( {24 + 1} \right)} \right)^{\dfrac{3}{4}}} \\
\Rightarrow{\left( {625} \right)^{\dfrac{3}{4}}} = {\left( {25 \times 25} \right)^{\dfrac{3}{4}}} \\
\Rightarrow{\left( {625} \right)^{\dfrac{3}{4}}} = {\left( {{{\left( {25} \right)}^2}} \right)^{\dfrac{3}{4}}}..........................\left( v \right) \\ \]
Now in (v) we can apply the exponential property ${\left( {{x^a}} \right)^b} = {x^{ab}}$.So we get:
\[{\left( {625} \right)^{\dfrac{3}{4}}} = {\left( {{{\left( {25} \right)}^2}} \right)^{\dfrac{3}{4}}} \\
\Rightarrow {\left( {625} \right)^{\dfrac{3}{4}}} = {\left( {25} \right)^{2 \times }}^{\left( {\dfrac{3}{4}} \right)} \\
\Rightarrow {\left( {625} \right)^{\dfrac{3}{4}}} = {\left( {25} \right)^{\dfrac{6}{4}}} \\
\Rightarrow {\left( {625} \right)^{\dfrac{3}{4}}} = {\left( {25} \right)^{\dfrac{3}{2}}}......................\left( {vi} \right) \\ \]
Now to simplify or to solve the given question let’s express $25$ as ${5^2}$ since then there is more chance of cancellation.So we can write:
\[{\left( {625} \right)^{\dfrac{3}{4}}} = {\left( {25} \right)^{\dfrac{3}{2}}} = {\left( {{5^2}} \right)^{\dfrac{3}{2}}}\]
Now again apply the exponential property ${\left( {{x^a}} \right)^b} = {x^{ab}}$, such that:
\[{\left( {625} \right)^{\dfrac{3}{4}}} = {\left( 5 \right)^{2 \times \dfrac{3}{2}}} \\
\Rightarrow{\left( {625} \right)^{\dfrac{3}{4}}}= {\left( 5 \right)^3}................................\left( {vii} \right) \\ \]
Now by simply solving (vii) we can write the final answer. Such that:
\[{\left( {625} \right)^{\dfrac{3}{4}}} = {\left( 5 \right)^3} = 125..................\left( {viii} \right)\]

Therefore ${\left( {625} \right)^{\dfrac{3}{4}}}$ in radical form can be written as $125$.

Note:The above problem can also be solved directly by using the formula ${a^{\dfrac{x}{n}}} = \sqrt[n]{{{a^x}}}\;{\text{where }}n{\text{ is a positive integer greater than }}x{\text{ and }}a{\text{ is a real number}}$. Also while solving this type of question care must be given while taking the root or while applying the exponential formulas. The easiest way to get a solution is always preferable since it saves a lot of time.