Question

Please help me with the 9th qn :)

Answer 1

$\underline{\underline{\Huge{\textbf{Question}}}}$

$\sf\textsf{Prove that}\\\large{\mathsf{\dfrac{\left(a^{p+q}\right)^{2}\, \left(a^{q+r}\right)^{2}\, \left(a^{r+p}\right)^{2}}{\left(a^{p}\cdot a^{q}\cdot a^{r}\right)^{4}}}}$





$\underline{\underline{\Huge{\textbf{Solution}}}}$

$\underline{\LARGE{\textsf{Step-1:}}}$

$\sf\textsf{Consider LHS of Eq. [A]}$

$\Large{\underline{\textbf{LHS \: =}}}$

$\mathsf{\dfrac{\left(a^{p+q}\right)^{2}\, \left(a^{q+r}\right)^{2}\, \left(a^{r+p}\right)^{2}}{\left(a^{p}\cdot a^{q}\cdot a^{r}\right)^{4}}}$ ———[A]





$\underline{\LARGE{\textsf{Step-2:}}}$

$\sf\textsf{Expand each term of Numerator of [A] using the Identity}$

$\qquad\LARGE{\boxed{\quad\bigstar\quad \mathbf{(a^{m})^{n}= a^{mn}\quad}}}$

$\begin{aligned} \sf \left(a^{p+q}\right)^{2}\, & = \sf a^{2(p+q)}& \implies \sf a^{2p+2q} \\\\\ \sf \left(a^{q+r}\right)^{2}\, & = \sf a^{2(q+r)}&\implies \sf a^{2q+2r} \\ \\ \sf \left(a^{r+p}\right)^{2}\, &= \sf a^{2(r+p)}&\implies \sf a^{2r+2p}\\\end{aligned}$





$\underline{\LARGE{\textsf{Step-3:}}}$

$\sf\textsf{Substitute the expanded terms in [A]}$

$\implies \mathsf{\dfrac{a^{2p+2q}\cdot a^{2q+2r}\cdot a^{2r+2p}}{\left(a^{p}\cdot a^{q}\cdot a^{r}\right)^{4}}}$ ———[1]





$\underline{\LARGE{\textsf{Step-4:}}}$

$\sf\textsf{Expand each term of Numerator and Denominator}\\\sf\textsf{of Eq.[1] using the Identity}$

$\qquad\LARGE{\boxed{\quad\bigstar\quad \mathbf{a^{m} \cdot a^{n}= a^{m+n}\quad}}}$

$\begin{aligned} \sf a^{2p+2q}\cdot a^{2q+2r}\cdot a^{2r+2p} &= \sf a^{2p+2q+2q+2r+2r+2p} \\ &= \sf a^{4p+4q+4r}\\\\ \sf a^{p}\cdot a^{q}\cdot a^{r} &= \sf a^{p+q+r}&\end{aligned}$





$\underline{\LARGE{\textsf{Step-5:}}}$

$\sf\textsf{Substitute the expanded terms in eq. [1]}$

$\implies \mathsf{\dfrac{a^{4p+4q+4r}}{\left(a^{p+q+r}\right)^{4}}}$ ———[2]





$\underline{\LARGE{\textsf{Step-6:}}}$

$\sf\textsf{Expand Denominator of Eq[2] using the Identity}$

$\qquad\LARGE{\boxed{\quad\bigstar\quad \mathbf{(a^{m})^{n}= a^{mn}\quad}}}$

$\begin{aligned}\sf \left(a^{p+q+r}\right)^{4} &= \sf a^{4(p+q+r)}\\&= \sf a^{4p+4q+4r}\end{aligned}$





$\underline{\LARGE{\textsf{Step-7:}}}$

$\sf\textsf{Substitute the expanded terms in eq. [2]}$

$\implies \mathsf{\dfrac{\cancel{a^{4p+4q+4r}}}{\cancel{a^{4p+4q+4r}}}}$

$\implies 1$

$\Large{\underline{\textbf{=\: RHS}}}$





$\huge{\underline{\LARGE{\textbf{Hence\: Proved}}}}$