Question

show that (exponents)​

Answer 1

${ \mathtt{\large{ \underline{Concept}}}}$

✏️Here the concept of Laws of exponents is used.

✏️The bases are same with different exponent powers which are multiplied. So in this case the the powers get added.

It can be written as :-

$\sf {\implies {x}^{y}\times {x}^{m}= {x}^{y + m} }$

✏️Any variable or constant raised to the power 0 is always 1.

example - a⁰= 1

${ \mathtt{\large{ \underline{Solution}}}}$

L.H.S

$\sf{ \sqrt{ {x}^{(p - q)} } \times \sqrt{ {x}^{(q - r)}} \times \sqrt{ {x}^{(r - p)}}} \\ \\ \sf{ \implies \: {x}^{ \frac{p - q}{2} } \times {x}^{ \frac{q - r}{2} } \times {x}^{ \frac{r - p}{2} } \: \: \: \: \: \: \: \: \: \: \: \: \: [ \because√=\frac{1}{2}]} \\ \\ \sf{ \implies \: {x}^{ \frac{p - q}{2} + \frac{q - r}{2} + \frac{r - p}{2}}} \\ \mathtt { \underline\color{lightblue}{add \: the \: powers}} \\ \\\sf{ \implies{x}^{ \frac{0}{2} } } \\ \\ \sf \implies\color{red}{1}$

R.H.S

↝ 1

Hence proved that L.H.S = R.H.S =1

_______________________

⭐Refer to the attachment for more rules of exponent!!

Answer 2

${\mathtt{\large{\underline {Concept}}}}$

✏️Here the concept of Laws of exponents is used.

✏️The bases are same with different exponent powers which are multiplied. So in this case the the powers get added

It can be written as :-

$\begin{gathered}\sf {\implies {x}^{y}\times {x}^{m}= {x}^{y + m} } \\ \end{gathered}$

✏️Any variable or constant raised to the power 0 is always 1.

example - a⁰= 1

${\mathtt{\large{\underline {Solution}}}}$

L.H.S

$\begin{gathered}\sf{ \sqrt{ {x}^{(p - q)} } \times \sqrt{ {x}^{(q - r)}} \times \sqrt{ {x}^{(r - p)}}} \\ \\ \sf{ \implies \: {x}^{ \frac{p - q}{2} } \times {x}^{ \frac{q - r}{2} } \times {x}^{ \frac{r - p}{2} } \: \: \: \: \: \: \: \: \: \: \: \: \: [ \because√=\frac{1}{2}]} \\ \\ \sf{ \implies \: {x}^{ \frac{p - q}{2} + \frac{q - r}{2} + \frac{r - p}{2}}} \\ \mathtt { \underline\color{purple}{add \: the \: powers}} \\ \\\sf{ \implies{x}^{ \frac{0}{2} } } \\ \\ \sf \implies\color{red} {1}\end{gathered}$

R.H.S

↝ 1

Hence proved that L.H.S = R.H.S =1