Question: If \(9(^{x-\frac{1}{2}}) –2^{(2x–2)} = 4^x – 3^{(2x–3)}\), then what is the value of x ?

1. 3/2
2. 3/4
3. 4/9
4. 2/5
5. 1/5

Correct Answer: A
Solution and Explanation:
Approach Solution 1:

It is given that 9^(x−1/2) –2^(2x–2) = 4^x – 3^(2x–3), we need to find the value of x.

By rearranging the equation, we get:9^(x−1/2) + 3^(2x–3) = 4^x + 2^(2x–2)
Or, (3^2)^(x−1/2) +3^(2x–3) =(2^2)^x + 2^(2x–2)

By applying the Power of a Power law, we get:
3^(2x−1) + 3^(2x–3) = 2^2x + 2^ (2x–2)

By factorising each side, we get:
3^(2x–3) (3^2+1) =2^(2x–2)(2^2+1)

By evaluating, we get:
3^(2x–3) (10)= 2^(2x–2) (5)
By dividing both sides by 10, we get:

3^(2x–3) = [2^(2x–2) (5)] /10
=> 3^(2x–3) = 2^(2x–2)/ 2
=> 3^(2x–3) = 2^(2x–3)

By dividing both sides by 2^(2x–3) we get:
3^(2x–3)/2^(2x–3) =1
=>(3/2)^(2x–3) =1
Therefore, we can say, 2x−3=0
Hence, the value of x= 3/2

Approach Solution 2:

The problem statement states that:

Given:

• 9^(x−1/2) –2^(2x–2) = 4^x – 3^(2x–3)

Find out:

• The value of x.

We can solve the problem by analysing the options.

9^(x−1/2) − 2^(2x−2) = 4^x− 3^(2x−3) can be derived as 3^(2x−1)−2^(2x−2) = 2^(2x)−3^(2x−3)

When analyzing the powers we notice that x is multiplied by 2. By comparing the left-hand side to the right-hand side, we can infer that x should be a multiple of 0.5, so that an integer comes in the power.

By exploring all the options, we find only 3/2 = 1.5 satisfies this.
By substituting x = 1.5, we get:
=>3^(3−1) − 2^(3−2) = 2^3 − 3^(3−3)
=>3^2 − 2^1 = 2^3 − 3^0

LHS = 9 - 2 = 7 and RHS = 8 - 1 = 7

There are no other values of x that would have offered us integer values in the power and would enable this comparison.

Approach Solution 3:

The problem statement states that:

Given:

• 9^(x−1/2) –2^(2x–2) = 4^x – 3^(2x–3)

Find out:

• The value of x.

The equation 9^(x−1/2) –2^(2x–2) = 4^x – 3^(2x–3) can be rewritten as:

=>3^2x/3 - 2^2x/4 = 2^2x - 3^2x/27
=> 10/27 x 3^2x = 5/4 x 2^2x
=> 3^2x/27 = 2^2x/8
=> (3/2)^2x = 27/8 = (3/2)^3
=>(3/2)^2x = (3/2)^3
=>2x = 3
=> x = 3/2

“If 9^(x−1/2) –2^(2x–2) = 4^x – 3^(2x–3), then what is the value”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “GMAT Official Guide 2019”. The GMAT Problem Solving questions measure the candidates’ abilities in calculating the numerical problems. GMAT Quant practice papers help the candidates to get acquainted with varieties of questions that will enhance their learning.

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