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Simply the expression $\dfrac{{{x^{\dfrac{3}{2}}}}}{{\dfrac{3}{2}}}$ and elaborate the steps.

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Last updated date: 23rd Feb 2024
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IVSAT 2024
Answer
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Hint:First step in simplifying the question is that any variable when divided by a fraction implies that the variable is multiplied by the reciprocal of the fraction. Next, we simplify the exponent of the numerator of the overall fraction by breaking the exponent into parts. The final step will be rearranging the expression.

Complete step by step solution:
As we know that the division of any number or any expression by a fraction equals to the
multiplication of the same expression by the reciprocal of the fraction. The expression in the question can be written as
$\dfrac{{{x^{\dfrac{3}{2}}}}}{{\dfrac{3}{2}}} = {x^{\dfrac{3}{2}}} \times (\dfrac{2}{3}) =
\dfrac{{2{x^{\dfrac{3}{2}}}}}{3}$
Let us now simplify the variable part of the expression, that is ${x^{\dfrac{3}{2}}}$.
We know that, ${x^{\dfrac{1}{2}}} = \sqrt x $.
So, ${x^{\dfrac{3}{2}}} = {({x^{\dfrac{1}{2}}})^3} = {({x^3})^{\dfrac{1}{2}}} = \sqrt {{x^3}} $… (Let this be equation (i))
Now that we have simplified the numerator of the expression, let us put the value of equation (i) in the place of the variable to arrive at the final expression.
$\dfrac{{{x^{\dfrac{3}{2}}}}}{{\dfrac{3}{2}}} = \dfrac{{2{x^{\dfrac{3}{2}}}}}{3} = \dfrac{{2\sqrt {{x^3}} }}{3}$

Note: In this question we may also simplify ${x^{\dfrac{3}{2}}} = {x^{\dfrac{1}{2} + 1}} = {x^{\dfrac{1}{2}}} \times x$. If we put this in the expression, we get $\dfrac{{{x^{\dfrac{3}{2}}}}}{{\dfrac{3}{2}}} = \dfrac{{2{x^{\dfrac{1}{2}}} \times x}}{3}$. This can be further written as, $\dfrac{{2x\sqrt x }}{3}$. In these types of simplifications, it is all subjective to when one wants to stop simplifying. There are also types of expressions that cannot be simplified, but rearranged. Some questions can just be a mixture of both.