If $$(0.064)×(0.4)^7=(0.4)^x×(0.0256)^2 ,$$​ then the value of $$\frac{x^{-1}-1}{x-1}$$​ is

Correct option is A

Given:
​$$(0.064) × (0.4)^7 = (0.4)^x × (0.0256)^2$$​​
Formula Used:
Laws of indices: $$a^m × a^n = a^{m+n}$$​ and $$(a^m)^n = a^{m × n}$$​​
Solution:
Express all base numbers as powers of 0.4.
​$$0.064 = (0.4)^3$$​​
​$$0.0256 = (0.4)^4$$​​
Substitute these into the original equation:
​$$(0.4)^3 × (0.4)^7 = (0.4)^x × ((0.4)^4)^2$$​​
Add exponents on the left and multiply on the right.
​$$(0.4)^{3+7} = (0.4)^x × (0.4)^8$$​​
​$$(0.4)^{10} = (0.4)^{x+8}$$​​
Equating the powers since the bases are identical:
10 = x + 8
x = 2
Now, substitute x = 2 into the required expression $$\frac{x^{-1} - 1}{x - 1}.$$​​
Expression = $$\frac{2^{-1} - 1}{2 - 1} = \frac{0.5 - 1}{1} = -0.5 = -\frac{1}{2}$$​​
Final Answer
So the correct answer is (a)