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We need to determine whether 4^(x+y) = 8^10. We start by breaking down our two bases into prime factors.

4^(x+y) = (2^2)^(x+y) = 2^(2x+2y)

8^10 = (2^3)^10 = 2^30

We can now rephrase the question as:

Is 2^(2x+2y) = 2^30 ?

Because the bases are the same, we can drop them and set the exponents equal to each other. The question becomes:

Is 2x+2y = 30 ?

Is x + y = 15 ?

After simplifying the equation, we see that we need to determine whether the sum of x and y is equal to 15.

Statement One Alone:

x – y = 9

Knowing the difference of x and y is not the same as knowing the sum of x and y; thus, statement one is not sufficient to answer the question. We can eliminate answer choices A and D.

Statement Two Alone:

y/x = ¼

When we cross multiply obtain:

4y = x

4y = x is not enough information to determine the value of x + y. Statement two alone is not sufficient. We can eliminate answer choice B.

Statements One and Two Together:

Using statements one and two we know the following:

x – y = 9 and 4y = x

Since 4y = x, we can substitute 4y for x into the equation x – y = 9 and we have:

4y – y = 9

3y = 9

y = 3

Since y = 3, x = 4(3) = 12.

Thus, x + y = 12 + 3 = 15. We can answer yes to the question. Both statements together are sufficient.

The answer is C.
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Clearly not sufficient as we can have multiple values for x and y

Hence, (1) ===== is NOT SUFFICIENT

(2) \frac{y}{x = 1/4}

\(x = 4y\)

Again this is not sufficient as we can get multiple values of x and y

Hence, (2) ===== is NOT SUFFICIENT

Combining (1) & (2)

When we substitute \(x = 4y\) in equation \(x - y = 9\)

we get,

\(4y - y = 9\)

\(3y = 9\)

\(y = 3\)

\(x = 4y\)

\(x = 4 * 3\)

\(x = 12\)

\(x + y = 12 + 3 = 15\)

Hence, (1) & (2) ===== is SUFFICIENT

Hence, Answer is C

Did you like the answer? 1 Kudos Please _________________

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Target question:Is 4^(x + y) = 8^10? This is a good candidate for rephrasing the target question.

Given equation: 4^(x + y) = 8^10 Rewrite 4 and 8 as powers of 2 to get: (2²)^(x + y) = (2³)^10 Apply power of a power law to get: 2^(2x +2y) = 2^30 This means that: 2x + 2y = 30 Divide both sides by 2 to get: x + y = 15 In other words, asking whether 4^(x + y) = 8^10 is the SAME as asking whether x + y = 15 REPHRASED target question:Is x + y = 15?

Statement 1: x - y = 9 Is this enough information to answer the REPHRASED target question? No. Consider these two CONFLICTING cases: Case a: x = 12 and y = 3. In this case, x + y = 12 + 3 = 15. So, x + y DOES equal 15 Case b: x = 10 and y = 1. In this case, x + y = 10 + 1 = 11. So, x + y does NOT equal 15 Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: y/x = 1/4 Is this enough information to answer the REPHRASED target question? No. Consider these two CONFLICTING cases: Case a: x = 12 and y = 3. In this case, x + y = 12 + 3 = 15. So, x + y DOES equal 15 Case b: x = 8 and y = 2. In this case, x + y = 8 + 2 = 10. So, x + y does NOT equal 15 Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined Statement 1 tells us that x - y = 9 Statement 2 tells us that y/x = 1/4 Since we have 2 different linear equations with 2 variables, we COULD solve the system for the individual values of x and y, which means we COULD answer the REPHRASED target question with certainty. Of course, we wouldn't waste precious time performing such calculations, since our sole goal is to determine the sufficiency of the combined statements. Since we can answer the target question with certainty, the combined statements are SUFFICIENT

I understand that statement one alone is not sufficient, however when we plug in for statement 2 we end up with 4y=x. So wouldn’t it be as follows, 4y+y= 15 5y= 15 Y= 3 and then we can solve for x Isn’t the answer B instead?

I understand that statement one alone is not sufficient, however when we plug in for statement 2 we end up with 4y=x. So wouldn’t it be as follows, 4y+y= 15 5y= 15 Y= 3 and then we can solve for x Isn’t the answer B instead?

Hello

It is NOT given that 4^(x+y) = 8^10. We have to determine whether this is actually true or not.

Quick question from my side. Why are we allowed in Statement B to make this move:

y/x=1/4 -> 4y = x

I am confused, because we don`t know for sure that either x or y are really positive and hence we would not be allowed to divide or multiple a variable.

Quick question from my side. Why are we allowed in Statement B to make this move:

y/x=1/4 -> 4y = x

I am confused, because we don`t know for sure that either x or y are really positive and hence we would not be allowed to divide or multiple a variable.

Thank you for your help!

We are concerned about the sign when dealing with inequalities: we should keep the sign if we multiply by a positive value and flip the sign when we multiply by a negative value.

For equations we can multiply/divide by a variable regardless of its sign (providing it's not 0).