Question

This is from the lesson exponents. Please help me...​

Answer 1

Given :-

• $\tt\small{ \left(\dfrac57\right)^{2n} \times\left\{\left( \dfrac{49}{25} \right)\div\left(\dfrac75\right)^{3} \right\} = \left( \dfrac{5}{7} \right)^{ - 3} }$

To find : Value of n

Solution :-

$\tt\small{ \implies\left(\dfrac57\right)^{2n} \times\left\{\left( \dfrac{49}{25} \right)\div\left(\dfrac75\right)^{3} \right\} = \left( \dfrac{5}{7} \right)^{ - 3} }$

$\tt\small{ \implies\left(\dfrac57\right)^{2n} \times\left\{\left( \dfrac{49}{25} \right) \times \left(\dfrac57\right)^{3} \right\} = \left( \dfrac{7}{5} \right)^{3} }$

$\tt\small{ \implies\left(\dfrac57\right)^{2n} \times\left\{\left( \dfrac{49}{25} \right) \times \left(\dfrac{125}{343}\right) \right\} = \left( \dfrac{343}{125} \right) }$

$\tt\small{ \implies\left(\dfrac57\right)^{2n} \times\left\{\left( \dfrac{5}{7} \right) \right\} = \left( \dfrac{343}{125} \right) }$

$\tt\small{ \implies\left(\dfrac57\right)^{2n}= \left( \dfrac{343}{125} \right) \div \left( \dfrac{7}{5} \right) }$

$\tt\small{ \implies\left(\dfrac57\right)^{2n}= \left( \dfrac{343}{125} \right) \times \left( \dfrac{5}{7} \right) }$

$\tt\small{ \implies\left(\dfrac57\right)^{2n}= \left( \dfrac{49}{25} \right) }$

$\tt\small{ \implies\left(\dfrac57\right)^{2n}= \left( \dfrac{ {7}^{2} }{ {5}^{2} } \right) }$

$\tt\small{ \implies\left(\dfrac57\right)^{2n}= \left( \dfrac{ {7}}{ {5} } \right)^{2} }$

$\tt\small{ \implies\left(\dfrac57\right)^{2n}= \left( \dfrac{5}{7} \right)^{ - 2} }$

$\tt\small{ \implies 2n = - 2}$

$\tt\small{ \implies n = - \dfrac{2}{2} }$

$\boxed{ \tt\implies n = - 1}$

Therefore the required value of n is -1.