Calculate: ((7-3x)^{\frac{1)/(2)}+(3)/(2)x * (7-3x)^{-(1)/(2)}}{7-3x}

Expression: $\frac{ {\left( 7-3x \right)}^{\frac{ 1 }{ 2 }}+\frac{ 3 }{ 2 }x \times {\left( 7-3x \right)}^{-\frac{ 1 }{ 2 }} }{ 7-3x }$

Calculate the product

$\frac{ {\left( 7-3x \right)}^{\frac{ 1 }{ 2 }}+\frac{ 3x \times {\left( 7-3x \right)}^{-\frac{ 1 }{ 2 }} }{ 2 } }{ 7-3x }$

If a negative exponent is in the numerator, move the expression to the denominator and make the exponent positive

$\frac{ {\left( 7-3x \right)}^{\frac{ 1 }{ 2 }}+\frac{ 3x }{ 2{\left( 7-3x \right)}^{\frac{ 1 }{ 2 }} } }{ 7-3x }$

Write all numerators above the common denominator

$\frac{ \frac{ 2\left( 7-3x \right)+3x }{ 2{\left( 7-3x \right)}^{\frac{ 1 }{ 2 }} } }{ 7-3x }$

Distribute $2$ through the parentheses

$\frac{ \frac{ 14-6x+3x }{ 2{\left( 7-3x \right)}^{\frac{ 1 }{ 2 }} } }{ 7-3x }$

Collect like terms

$\frac{ \frac{ 14-3x }{ 2{\left( 7-3x \right)}^{\frac{ 1 }{ 2 }} } }{ 7-3x }$

Simplify the complex fraction

$\frac{ 14-3x }{ 2{\left( 7-3x \right)}^{\frac{ 1 }{ 2 }} \times \left( 7-3x \right) }$

Use ${a}^{\frac{ m }{ n }}=\sqrt[n]{{a}^{m}}$ to transform the expression

$\frac{ 14-3x }{ 2\sqrt{ 7-3x }\left( 7-3x \right) }$