Question

then what will be the value of x plz answer

Answer 1

This is a very tricky question and requires the knowledge of basic mathematical concepts & identities. It also requires the ability to identify certain mathematical values or calculate them as quickly as possible. The question may look lengthy, but it actually requires the application of basic mathematical knowledge.This question took me about 5 minutes to analyse and solve, but a competitive exam paper demands more speed but with the same amount of precision and accuracy. You can solve this question in various methods, one of which I will explain in detail below. I will also explain how to solve such questions along the way.

First, we notice that in the number $13824^{2/3}$, there is 3 in the denominator of the power. Thus, the number must be a perfect cube of some other number (or else the problem would have become too lengthy to solve for a competitive paper).
We can see that there is a number 24 in the denominator of the LHS. On observation, we come to know that $24^{3}$ = 13824. Thus, the RHS becomes $( 24^{3})^{2/3}$ = $24^{2}$.

Moving on to the LHS, we know that $\sqrt{x}$ can be written as $x^{1/2}$. We have four square roots in the numerator. It can be re-written as:
 $( ( ( ( 5^{x-1} - 7^{x-3} ) ^{1/2} ) ^{1/2} ) ^{1/2} ) ^{1/2}$ = $( 5^{x-1} - 7^{x-3} ) ^{1/16}$.

Coming to the denominator, we see that there is a whole number $24^{5/8}$. The denominator of the power usually indicates the order of the power, i.e. third root, fourth root, etc. Here, we have 8 in the denominator of the power. But, as such we do not know the 8th root of 24. Thus, we multiply the whole problem with the power 8 to eliminate the root.

Thus, the problem becomes:
$( 5^{x-1} - 7^{x-3} ) ^{8/16}$ / $( 8^{x-2} + 3^{x-1} - 17 ) ^{-40/2}$ * $24^{25}$ = $24^{16}$

Shifting the whole number to the RHS, we have:
$( 5^{x-1} - 7^{x-3} ) ^{1/2}$ / $( 8^{x-2} + 3^{x-1} - 17 ) ^{-20}$  = $24^{25+16}$

$( 5^{x-1} - 7^{x-3} ) ^{1/2}$ / $( 8^{x-2} + 3^{x-1} - 17 ) ^{-20}$  = $24^{41}$

The denominator has a negative power, thus we can shift it to the numerator.
$( 5^{x-1} - 7^{x-3} ) ^{1/2}$ * $( 8^{x-2} + 3^{x-1} - 17 ) ^{20}$  = $24^{41}$

Hence, the whole LHS must simplify and give us $24^{41}$. It takes a long time to solve this equation, thus we adopt the inspection method. In this method, we substitute the values given in the options and conclude which one satisfies the equation.

Here, I take x=5. Then, the LHS becomes:
$( 5^{5-1} - 7^{5-3} ) ^{1/2}$ * $( 8^{5-2} + 3^{5-1} - 17 ) ^{20}$
= $( 5^{4} - 7^{2} ) ^{1/2}$ * $( 8^{3} + 3^{4} - 17 ) ^{20}$
= $( 625 - 49 ) ^{1/2}$ * $( 512 + 81 - 17 ) ^{20}$
= $( 576 ) ^{1/2}$ * $( 576 ) ^{20}$ 

We know that $\sqrt{576}$ = 24 and 576 = $24^{2}$. Thus, we can further simplify it as

24 * $( 24^{2} ) ^{20}$
= 24 * $24^{40}$
= $24^{41}$
= RHS

Thus, we can conclude that the value of x is 5.
Hence, (C) is the correct option.