Question

If 27^k = 9/3^k, then the value of 1/k^2 is 1.) 4 2.)0 3.)Both(1) and (2) 4.) None of these Please explain the answer with steps....

Answer 1

4.none of these

${27}^{k} = { (\frac{9}{3} })^{k} \\ {3}^{3k} = {3}^{k} \\ 3k = k \\ 3k - k = 0 \\ 2k = 0 \\ k = 0$

${ \frac{1}{k} }^{2} = { \frac{1}{0} }^{2}$

Answer 2

Answer:

Required value is 4

So option 1) is the correct answer.

Step-by-step explanation:

Given,

${27}^{k} = \frac{9}{ {3}^{k} }$

${ {3}^{3} }^{k} = \frac{9}{ {3}^{k} }$

${3}^{3k} = \frac{9}{ {3}^{k} } \\ {3}^{3k} \times {3}^{k} = 9 \\ {3}^{3k + k} = 9 \\ {3}^{4k} = 9 \\ {3}^{4k} = {3}^{2} \\ 4k = 2 \\ k = \frac{2}{4} \\ k = \frac{1}{2}$

So,

$\frac{1}{ {k}^{2} } = \frac{1}{ { \frac{1}{2} }^{2} } = 4$

So required value is 4

Here applied formulas are

${ {x}^{a} }^{b} = {x}^{ab} \\ {x}^{a} \times {x}^{b} = {x}^{a + b}$