Question

Which of the following statements is CORRECT?
Statement I : If ${\left(\frac{5}{4}\right)}^5\times {\left(\frac{5}{4}\right)}^{11}={\left(\frac{5}{4}\right)}^{8x}$; then $x=2$
Statement II : If ${\left(\frac{2}{5}\right)}^3\times {\left(\frac{2}{5}\right)}^7={\left(\frac{2}{5}\right)}^{x-1}$; then $x=11$

• A. Only Statement I
• B. Only Statement II
• C. Both Statement I and Statement II
• D. Neither Statement I nor Statement II

Answer 1

The correct option is C Both Statement I and Statement II

Statement I :

${\left(\frac{5}{4}\right)}^5\times {\left(\frac{5}{4}\right)}^{11}={\left(\frac{5}{4}\right)}^{8x}$

$\Rightarrow {\left(\frac{5}{4}\right)}^{8x}={\left(\frac{5}{4}\right)}^{5+11}\Rightarrow {\left(\frac{5}{4}\right)}^{8x}={\left(\frac{5}{4}\right)}^{16}$

$\therefore 8x=16\Rightarrow x=2$

Statement II :

${\left(\frac{2}{5}\right)}^3\times {\left(\frac{2}{5}\right)}^7={\left(\frac{2}{5}\right)}^{x-1}$

$\Rightarrow {\left(\frac{2}{5}\right)}^{x-1}={\left(\frac{2}{5}\right)}^{3+7}\Rightarrow {\left(\frac{2}{5}\right)}^{x-1}={\left(\frac{2}{10}\right)}^{10}$

$\therefore x-1=10\Rightarrow x=11$

Hence, both statement I and statement II are correct.