For all positive integers w and y, where w > y, let the operation ☺ be defined by w ☺ y = (2^(w + y))/(2^(w - y)).
For how many positive integers w is w ☺ 1 equal to 4?
A. More than four
B. None
C. One
D. Two
E. Four
*I found only 3/5 as the answer but it was less than one so I picked B. It was wrong and the answer key said it was A. Why is that?
To find how many positive integers w satisfy w ☺ 1 = 4, we need to substitute w = 4 into the given expression and check if the equation holds true.
Using the given definition, we have:
4 ☺ 1 = (2^(4 + 1))/(2^(4 - 1))
= (2^5)/(2^3)
= 32/8
= 4
So, w = 4 is a valid solution.
Since the question asks for the number of positive integers w that satisfy the equation, we have found one solution. Thus, the correct answer is C. One.
To find the number of positive integers w for which w ☺ 1 is equal to 4, we need to substitute the given expression into the equation and solve for w.
Starting with the expression w ☺ y = (2^(w + y))/(2^(w - y)), we substitute 1 for y and 4 for the result to get:
w ☺ 1 = (2^(w + 1))/(2^(w - 1)) = 4
Now we simplify the equation:
(2^(w + 1))/(2^(w - 1)) = 4
Next, let's apply the properties of exponents, in particular, the rule that states that dividing two numbers with the same base is equivalent to subtracting their exponents:
2^(w + 1 - (w - 1)) = 4
Simplifying further:
2^(2) = 4
Since 2^2 = 4, the equation simplifies to:
4 = 4
This equation is true for all positive integers w.
Now, referring back to the answer choices:
A. More than four
B. None
C. One
D. Two
E. Four
From our equation, we can see that the number of possible values for w that satisfy the equation is infinite. Therefore, the correct answer is A. More than four.
It seems you made a mistake by assuming that the equation had a finite number of solutions. Due to the nature of the equation, the number of positive integers w that satisfy the equation is not limited to a specific value or range.