Question

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Answer 1

When I see this question, I wished to find the value of x in a different 'zigzag' way!!! I like to do as those too!!!

Here, in $2^{x-2} \cdot 3^{2x-6}=36$,

it seems that 36 is subjected to prime factorization.

And also it is true that 36 has only two prime factors, i.e., 2 and 3.

So the first method which came in my mind was, to find it by the no. of factors of 36!!!

Yes, the no. of factors of $2^{x-2} \cdot 3^{2x-6}=36$ should be equal to that of 36.

The method to find the no. of factors of a number is to find the product of the numbers obtained by adding 1 to each of the exponents of the primes in prime factorization.

Let me get it in your mind.

36 = 2² × 3²

Here, exponent of both 2 and 3 is 2 each.

Add 1 to each and then multiply.

(2 + 1)(2 + 1) = 3 × 3 = 9

∴ 36 has 9 factors. So has $2^{x-2} \cdot 3^{2x-6}=36$.

Here, exponents of 2 and 3 are $x-2$ and $2x-6$ respectively.

So add 1 to each and multiply both, which equals 9.

Therefore,

$(x-2+1)(2x-6+1)=9 \\ \\ (x-1)(2x-5)=9 \\ \\ 2x^2-5x-2x+5=9 \\ \\ 2x^2-7x+5=9 \\ \\ 2x^2-7x+5-9=0 \\ \\ 2x^2-7x-4=0 \\ \\ 2x^2+x-8x-4=0 \\ \\ x(2x+1)-4(2x+1)=0 \\ \\ (x-4)(2x+1)=0 \\ \\ \\ \therefore\ x=4\ \ \ ; \ \ \ x=-\frac{1}{2}$

$x=-\frac{1}{2}$ is not possible because it changes RHS of $2^{x-2} \cdot 3^{2x-6}=36$, i.e., 36.

∴ x = 4.

So I found the value of x using a different way.

I'm sure less people would use such method, and most of the people do the following:

$2^{x-2} \cdot 3^{2x-6}=36 \\ \\ 2^{x-2} \cdot 3^{2(x-3)}=4 \cdot 9 \\ \\ 2^{x-2} \cdot (3^2)^{(x-3)}=2^2 \cdot 9^1 \\ \\ 2^{x-2} \cdot 9^{x-3}=2^2 \cdot 9^1 \\ \\ \\ x-2=2 \\ \\ x=\bold{4} \\ \\ \\ OR \\ \\ \\ x-3=1 \\ \\ x=\bold{4}$

Hope this would also be helpful to you.

Please if you've any doubt, then ask me.

Also please mark it as the brainliest if this helps you.

Thank you. :-))