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If $x^{x \sqrt{x}} = {(x \sqrt{x})}^{x},$ ${x^{x\sqrt x }} = {\left( {x\sqrt x } \right)^x}{\text{,}}$ then x equals to?

1 $\frac{4}{9}$ $\frac{4}{9}$
2 $\frac{2}{3}$ $\frac{2}{3}$
3 $\frac{9}{4}$ $\frac{9}{4}$
4 $\frac{3}{2}$ $\frac{3}{2}$

Answer:- 1

Explanation:-

Solution:

\begin{aligned} x^{x \sqrt{x}} = {(x \sqrt{x})}^{x} \\ x^{x \sqrt{x}} = {(x^{\frac{3}{2}})}^{x} \\ x^{x \sqrt{x}} = x^{\frac{3}{2} x} \end{aligned}

$\eqalign{ & {x^{x\sqrt x }} = {\left( {x\sqrt x } \right)^x} \cr & {x^{x\sqrt x }} = {\left( {{x^{\frac{3}{2}}}} \right)^x} \cr & {x^{x\sqrt x }} = {x^{\frac{3}{2}x}} \cr}$
(If bases are same then their power is also same)

\begin{aligned} ∴ x \sqrt{x} = \frac{3}{2} x \\ \Rightarrow \sqrt{x} = \frac{3}{2} \\ \Rightarrow x = {(\frac{3}{2})}^{2} \\ \Rightarrow x = \frac{9}{4} \end{aligned}

$\eqalign{ & \therefore x\sqrt x = \frac{3}{2}x \cr & \Rightarrow \sqrt x = \frac{3}{2} \cr & \Rightarrow x = {\left( {\frac{3}{2}} \right)^2} \cr & \Rightarrow x = \frac{9}{4} \cr}$

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then x equals to?", "text": "If

x^{x \sqrt{x}} = {(x \sqrt{x})}^{x},

${x^{x\sqrt x }} = {\left( {x\sqrt x } \right)^x}{\text{,}}$ then x equals to?", "dateCreated": "7/24/2019 10:09:12 AM", "author": { "@type": "Person", "name": "Nitin Sir" }, "answerCount": "4", "acceptedAnswer": { "@type": "Answer", "text": "

Solution:

\begin{aligned} x^{x \sqrt{x}} = {(x \sqrt{x})}^{x} \\ x^{x \sqrt{x}} = {(x^{\frac{3}{2}})}^{x} \\ x^{x \sqrt{x}} = x^{\frac{3}{2} x} \end{aligned}

$\eqalign{ & {x^{x\sqrt x }} = {\left( {x\sqrt x } \right)^x} \cr & {x^{x\sqrt x }} = {\left( {{x^{\frac{3}{2}}}} \right)^x} \cr & {x^{x\sqrt x }} = {x^{\frac{3}{2}x}} \cr}$
(If bases are same then their power is also same)

\begin{aligned} ∴ x \sqrt{x} = \frac{3}{2} x \\ \Rightarrow \sqrt{x} = \frac{3}{2} \\ \Rightarrow x = {(\frac{3}{2})}^{2} \\ \Rightarrow x = \frac{9}{4} \end{aligned}

$\eqalign{ & \therefore x\sqrt x = \frac{3}{2}x \cr & \Rightarrow \sqrt x = \frac{3}{2} \cr & \Rightarrow x = {\left( {\frac{3}{2}} \right)^2} \cr & \Rightarrow x = \frac{9}{4} \cr}$ ", "dateCreated": "7/24/2019 10:09:12 AM", "author": { "@type": "Person", "name": "Nitin Sir" } }, "suggestedAnswer": { "@type": "Answer", "text": "

Solution:

\begin{aligned} x^{x \sqrt{x}} = {(x \sqrt{x})}^{x} \\ x^{x \sqrt{x}} = {(x^{\frac{3}{2}})}^{x} \\ x^{x \sqrt{x}} = x^{\frac{3}{2} x} \end{aligned}

$\eqalign{ & {x^{x\sqrt x }} = {\left( {x\sqrt x } \right)^x} \cr & {x^{x\sqrt x }} = {\left( {{x^{\frac{3}{2}}}} \right)^x} \cr & {x^{x\sqrt x }} = {x^{\frac{3}{2}x}} \cr}$
(If bases are same then their power is also same)

\begin{aligned} ∴ x \sqrt{x} = \frac{3}{2} x \\ \Rightarrow \sqrt{x} = \frac{3}{2} \\ \Rightarrow x = {(\frac{3}{2})}^{2} \\ \Rightarrow x = \frac{9}{4} \end{aligned}

$\eqalign{ & \therefore x\sqrt x = \frac{3}{2}x \cr & \Rightarrow \sqrt x = \frac{3}{2} \cr & \Rightarrow x = {\left( {\frac{3}{2}} \right)^2} \cr & \Rightarrow x = \frac{9}{4} \cr}$ ", "dateCreated": "7/24/2019 10:09:12 AM" } }

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