Question

please give solution to the question in picture ​

Answer 1

$\rm \longrightarrow \large \bigg( \dfrac{ {x}^{p} }{ {x}^{q} } \bigg)^{p + q} \times \large \bigg( \dfrac{ {x}^{q} }{ {x}^{r} } \bigg)^{q + r} \times \large \bigg( \dfrac{ {x}^{r} }{ {x}^{p} } \bigg)^{ r + p}$

$\rm \longrightarrow \large \bigg( { {x}^{p - q} } \bigg)^{p + q} \times \large \bigg( { {x}^{q - r} } \bigg)^{q + r} \times \large \bigg( { {x}^{r - p} }{ } \bigg)^{ r + p}$

$\rm \longrightarrow \large \bigg( { {x} } \bigg)^{{(p - q)}(p + q)} \times \large \bigg( { {x}^{} } \bigg)^{(q - r)(q + r)} \times \large \bigg( { {x} }{ } \bigg)^{( r - p)(r + p)}$

$\rm \longrightarrow \large \bigg( { {x} } \bigg)^{{( {p}^{2} - {q}^{2} )}} \times \large \bigg( { {x}^{} } \bigg)^{( {q}^{2} - {r}^{2} )} \times \large \bigg( { {x} }{ } \bigg)^{( {r}^{2} - {p}^{2} )}$

$\rm \longrightarrow \large \bigg( { {x} } \bigg)^{{{( {p}^{2} - {q}^{2} )}} + {( {q}^{2} - {r}^{2} )} + {( {r}^{2} - {p}^{2} )}}$

$\rm \longrightarrow \large \bigg( { {x} } \bigg)^{{{( {p}^{2} - {q}^{2} }} + { {q}^{2} - {r}^{2} } + { {r}^{2} - {p}^{2} )}}$

$\rm \longrightarrow \large \bigg( { {x} } \bigg)^{{{( {p}^{2} - {p}^{2} + {q}^{2} }} - { {q}^{2} + {r}^{2} } - { {r}^{2} )}}$

$\rm \longrightarrow \large \bigg( { {x} } \bigg)^{0} =1$

Additional Information :-

$\begin{gathered}\boxed{\begin{minipage}{5 cm}\bf{\dag}\:\:\underline{\text{Law of Exponents :}}\\\\\bigstar\:\:\sf\dfrac{a^m}{a^n} = a^{m - n}\\\\\bigstar\:\:\sf{(a^m)^n = a^{mn}}\\\\\bigstar\:\:\sf(a^m)(a^n) = a^{m + n}\\\\\bigstar\:\:\sf\dfrac{1}{a^n} = a^{-n}\\\\\bigstar\:\:\sf\sqrt[\sf n]{\sf a} = (a)^{\dfrac{1}{n}}\end{minipage}}\end{gathered}$

Answer 2

Step-by-step explanation:

$\\ \\ \qquad \quad{\huge{\color {blue}{\sf\longmapsto \left(\dfrac {x^p}{x^q}\right)^{p+q}\times \left(\dfrac {x^q}{x^r}\right)^{q+p}\times \left(\dfrac {x^r}{x^p}\right)^{r+p}}}}$

$\\ \\ \qquad \quad\huge {\color {pink}{\sf\longmapsto \left (x^{p-q}\right)^{p+q}\times \left (x^{q-r}\right)^{q+r}\times \left (x^{r-p}\right)^{r+p}}}$

$\\ \\ \qquad \quad\huge{\color {purple}{ \sf\longmapsto x^{(p-q)(p+q)}\times x^{(q-r)(q+r)}\times x^{(r-p)(r+p)}}}$

$\\ \\ \qquad \quad\huge{\color {brown}{\sf\longmapsto x^{(p^2-q^2)}\times x^{(q^2-r^2)}\times x^{(r^2-p^2)}}}$

$\\ \\ \qquad \quad\huge{\color {darkblue}{\sf\longmapsto x^{\left\{(p^2-q^2)+(q^2-r^2)+(r^2-p^2)\right\}}}}$

$\\ \\ \qquad \quad\huge{\color{Orange}{\sf\longmapsto x^{\left\{p^2-q^2+q^2-r^2+r^2-p^2\right\}}}}$

$\\ \\ \qquad \quad\huge {\color {gray}{\sf\longmapsto x^{(0+0+0)}}}$

$\\ \\ \qquad \quad\huge{\color {lime}{\sf\longmapsto x^0}}$

$\\ \\ \qquad \quad\huge{\color {navy}{\sf\longmapsto 1}}$

Learn more:-

$\boxed{\begin{array}{c}\bf{\dag}\:\:\underline{\text{Law of Exponents :}}\\\\\bigstar\:\:\sf\dfrac{a^m}{a^n} = a^{m - n}\\\\\bigstar\:\:\sf{(a^m)^n = a^{mn}}\\\\\bigstar\:\:\sf(a^m)(a^n) = a^{m + n}\\\\\bigstar\:\:\sf\dfrac{1}{a^n} = a^{-n}\\\\\bigstar\:\:\sf\sqrt[\sf n]{\sf a} = (a)^{\dfrac{1}{n}}\end {array}}$