What is the value of 3^(-(x + y))/3^(-(x - y))?

\(\frac{3^{-(x + y)}}{3^{-(x - y)}}\)= \(3^{-(x + y)+(x - y)}\)=\(3^{-x+y+x-y}\)=\(3^{-2y}\). So the value will depend on the value of \(y\) only.

(1) x = 2. Not Sufficient

(2) y = 3. Sufficient

Answer B.

Solution

To find:
• The value of the expression \(\frac{3^{-(x + y)}}{3^{-(x - y)}}\)

Approach and Working:
We can simplify the given expression as

• \(\frac{3^{-(x + y)}}{3^{-(x - y)}}\)
= \(3^{(-x - y) – (-x + y)}\)
= \(3^{(-x - y + x - y)}\)
= \(3^{(- 2y)}\)

Therefore, if we know the value of y, we can find out the value of the expression

Analysing Statement 1

• As per the information given in statement 1, x = 2

o From this statement, we don’t find any information about y

Hence, statement 2 is not sufficient to answer

Analysing Statement 2

• As per the information given in statement 2, y = 3

o Substituting the value of y, we can find the value of the given expression

Hence, statement 2 is sufficient to answer

Hence, the correct answer is option B.

Answer: B
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The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.
The question asks the value of \(\frac{{3^{-(x+y)}}}{{3^{-(x-y)}}} = 3^{-(x+y)-(-(x-y))} = 3^{-x-y+x-y} = 3^{-2y}\).

Thus condition 2) is sufficient since it tells the value of y.

Condition 1)

Since we don’t know the value of x from condition 1), it does not yield a unique solution and condition 1) is not sufficient.

Therefore, B is the answer.

Target question: What is the value of \(\frac{3^{-(x + y)}}{3^{-(x - y)}}\)?
This is a good candidate for rephrasing the target question.

Take: \(\frac{3^{-(x + y)}}{3^{-(x - y)}}\)

Simplify both exponents: \(\frac{3^{(-x - y)}}{3^{(-x + y)}}\)

Apply the quotient law: \(3^{(-x - y) - (-x + y)}\)

Simplify: \(3^{-2y}\)

To find the value of \(3^{-2y}\), we only need to know the value of \(y\). So we can rephrase our target question...

REPHRASED target question: What is the value of \(y\)?

Aside: the video below has tips on rephrasing the target question

Statement 1: \(x = 2\)
No good. We need to find the value of \(y\).
NOT SUFFICIENT

Statement 2: \(y = 3\)
SUFFICIENT

Answer: B

VIDEO ON REPHRASING THE TARGET QUESTION: