Question

Simplify $\left( {\sqrt x } \right) \times x.$

Answer 1

Hint:We know that to multiply two terms having the same base, we can just add the exponents with each other. ${x^m} \times {x^n} = {\left( x \right)^{m + n}}$. In order to simplify the given expression we have to convert the given expression in the form of the above identity. After converting it to the above form we can just simplify it using simple arithmetic operations.

Complete step by step answer:
Given, $\left( {\sqrt x } \right) \times x.......................................\left( i \right)$
Normally to simplify the expressions containing square root we have to remove all perfect squares if any present inside the square root.But here we can see that there is no perfect root inside the square root such that to simplify it we have to use the expression:
${x^m} \times {x^n} = {\left( x \right)^{m + n}}.............................\left( {ii} \right)$
Such that we have to express the equation (i) in terms of (ii). We also know that:
$\sqrt x = {\left( x \right)^{\dfrac{1}{2}}}............................................\left( {iii} \right)$
So substituting (iii) in (i), we get:
$\left( {\sqrt x } \right) \times x = {\left( x \right)^{\dfrac{1}{2}}} \times x..............................\left( {iv} \right)$
Now we can see that (iv) and (ii) are similar expressions which is having same base such that we can apply the identity${x^m} \times {x^n} = {\left( x \right)^{m + n}}$.
Such that applying (ii) in (iv) we get:
$
{\left( x \right)^{\dfrac{1}{2}}} \times x = {\left( x \right)^{\left( {\dfrac{1}{2}} \right) + 1}} \\ \Rightarrow{\left( x \right)^{\dfrac{1}{2}}} \times x= {\left( x \right)^{\dfrac{{1 + 2}}{2}}} \\
\therefore{\left( x \right)^{\dfrac{1}{2}}} \times x = {\left( x \right)^{\dfrac{3}{2}}}...................................\left( v \right) \\
 $
Therefore on simplifying $\left( {\sqrt x } \right) \times x$ we get ${\left( x \right)^{\dfrac{3}{2}}}$.

Note:Whenever questions including exponents are given some of the identities useful are:
$
{x^m} \times {x^n} = {\left( x \right)^{m + n}} \\
\Rightarrow\dfrac{{{x^n}}}{{{x^m}}} = {\left( x \right)^{n - m}} \\
\Rightarrow{\left( {{x^n}} \right)^m} = {\left( x \right)^{n \times m}} \\ $
So our given expressions should be converted and expressed based on the above standard identities, by which it would be much easier to simplify and solve it. Also radical expressions are algebraic expressions which have or contain radicals, and the best way to solve a square root is to remove all the perfect squares from inside the square root if any exists.