a)Given:

\(x^{\frac{5}{2}}-9x^{\frac{1}{2}}\)

On simplification, we get

\(x^{\frac{5}{2}}-9x^{\frac{1}{2}}=x^{2}\cdot x^{\frac{1}{2}}-9x^{\frac{1}{2}} \begin{bmatrix}\because x^{m} \cdot x^{n} = x^{m+n} \\\Rightarrow x^{2} \cdot x^{\frac{1}{2}}=x^{2+\frac{1}{2}}=x^{\frac{5}{2}} \end{bmatrix}\)

\(=x^{\frac{1}{2}}(x^{2}-9)\)

\(= x^{\frac{1}{2}} (x^{2} - 3^{2})\)

\(=x^{\frac{1}{2}}(x-3)(x+3)[\because a^{2}-b^{2})=(a-b)(a+b)]\)

b)Given:

\(=x^{-\frac{3}{2}}+2x(-\frac{1}{2})+x^{\frac{1}{2}}ZSL

On simplification, we get

\(x^{-\frac{3}{2}}+2x(-\frac{1}{2})+x^{\frac{1}{2}}=x^{-\frac{3}{2}}+2x(-\frac{3}{2}) \cdot x + x^{\frac{3}{2}} \cdot x^{2}[(\because x^{m} \cdot x^{n}=x^{m+n}(\Rightarrow x^{-\frac{3}{2}} \cdot x=x^{-\frac{3}{2}+1=x^{-\frac{1}{2}}})\)

And \(x^{-\frac{3}{2}} \cdot x^{2}=x^{-\frac{3}{2}+2=x^{\frac{1}{2}}}\)

\(x^{\frac{3}{2}}(1+2x+x^{2})\)

\(=x^{-\frac{3}{2}}(1+x)^{2}[\because(1+x)^{2}=1+2x+x^{2}]\)

\(x^{\frac{5}{2}}-9x^{\frac{1}{2}}\)

On simplification, we get

\(x^{\frac{5}{2}}-9x^{\frac{1}{2}}=x^{2}\cdot x^{\frac{1}{2}}-9x^{\frac{1}{2}} \begin{bmatrix}\because x^{m} \cdot x^{n} = x^{m+n} \\\Rightarrow x^{2} \cdot x^{\frac{1}{2}}=x^{2+\frac{1}{2}}=x^{\frac{5}{2}} \end{bmatrix}\)

\(=x^{\frac{1}{2}}(x^{2}-9)\)

\(= x^{\frac{1}{2}} (x^{2} - 3^{2})\)

\(=x^{\frac{1}{2}}(x-3)(x+3)[\because a^{2}-b^{2})=(a-b)(a+b)]\)

b)Given:

\(=x^{-\frac{3}{2}}+2x(-\frac{1}{2})+x^{\frac{1}{2}}ZSL

On simplification, we get

\(x^{-\frac{3}{2}}+2x(-\frac{1}{2})+x^{\frac{1}{2}}=x^{-\frac{3}{2}}+2x(-\frac{3}{2}) \cdot x + x^{\frac{3}{2}} \cdot x^{2}[(\because x^{m} \cdot x^{n}=x^{m+n}(\Rightarrow x^{-\frac{3}{2}} \cdot x=x^{-\frac{3}{2}+1=x^{-\frac{1}{2}}})\)

And \(x^{-\frac{3}{2}} \cdot x^{2}=x^{-\frac{3}{2}+2=x^{\frac{1}{2}}}\)

\(x^{\frac{3}{2}}(1+2x+x^{2})\)

\(=x^{-\frac{3}{2}}(1+x)^{2}[\because(1+x)^{2}=1+2x+x^{2}]\)