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If x ≠ 0, what is the value of (x^w/x^y)^4?

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Source: — Data Sufficiency |
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by Vincen » Sat Mar 03, 2018 2:19 am
VJesus12 wrote:If x ≠ 0, what is the value of (x^w/x^y)^4?

(1) w = y

(2) x^2 = 1

The OA is D.

Experts, may you help me here? I need some help. I don't know how to prove that each statement alone is sufficient. <i class="em em-confused"></i>
Hello.

I think I can help you. We need to find the value of $$\left(\frac{x^w}{x^y}\right)^4$$ but this expression is equal to $$\left(x^{w-y}\right)^4=x^{4w-4y}$$ Now:

If w=y then 4w-4y=0 and then $$x^{4w-4y}=x^0=1.$$ (1) is sufficient.

On the other hand, if $$x^2=1\ \leftrightarrow\ \ \ x=\pm1$$ Now, no matter the sign of x, we have that $$\frac{x^w}{x^y}=\pm1\ \leftrightarrow\ \ \ \left(\pm1\right)^4=1$$ and therefore (2) is sufficient.

The correct answer is D .
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by Jay@ManhattanReview » Wed Mar 07, 2018 9:42 pm
VJesus12 wrote:If x ≠ 0, what is the value of (x^w/x^y)^4?

(1) w = y

(2) x^2 = 1

The OA is D.

Experts, may you help me here? I need some help. I don't know how to prove that each statement alone is sufficient. <i class="em em-confused"></i>
Given: x ≠ 0

We have to get the value of (x^w/x^y)^4.

Let's take each statement one by one.

(1) w = y

(x^w/x^y)^4 => (x^w/x^w)^4; replacing the value of w as y.

(x^w/x^w)^4 = (1/1)^4 = 1. Sufficient.

(2) x^2 = 1

=> x = ±1

Case 1: Say x = 1, then (x^w/x^y)^4 = (1^w/1^y)^4 = 1/1)^4 = 1.
Case 2: Say x = -1, then (x^w/x^y)^4 = ((-1)^w/(-1)^y)^4 = [±(1/1)]^4 = ±1^4 = 1. Sufficient.

Note that -1 raised to even integer is positive.

The correct answer: D

Hope this helps!

-Jay
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