For Integers x and y, 2^x+2^y=2^30. What is the Value of x+y? GMAT Problem Solving

Question: For integers x and y, 2^x+2^y=2^30. What is the value of x+y?

1. 30
2. 32
3. 46
4. 58
5. 64

Correct Answer: D

Solution ans Explanation
Approach Solution 1:
Given to us is an equation,
\(2^x\)+\(2^y\)=\(2^{30}\)
It is asked to find out the value of x + y.
This question is from the topic exponent and powers.
Given an equation,
\(2^x\)+ \(2^y\)= \(2^{30}\)
\(2^x\)(1 + \(2^{y-x}\)) = \(2^{30}\)
1+\(2^{y-x}\) = \(2^{30-x}\)
Difference between any two numbers which are the powers of 2 can only be one when the powers are 1 and 0.
I.e, \(2^1\)- \(2^0\)= 1
There is no other value that satisfies the equation other than 1 and 0.
Therefore coming to the equation,
\(2^{30-x}\) -\(2^{y-x}\) = 1
30 - x = 1
X = 29
Y - x = 0
Y = x = 29
The value of x + y = 29 + 29 = 58.
Therefore the correct answer is option D.

Approach Solution 2:
Given to us is an equation,
\(2^x\)+\(2^y\)=\(2^{30}\)
It is asked to find out the value of x + y.
This question is from the topic exponent and powers.
Given an equation,
\(2^x\)+\(2^y\)=\(2^{30}\)
There is a property of exponents which would help us in solving this problem.
\(2^1\)+ \(2^1\)= \(2^2\)
\(2^2\)+ \(2^2\)= \(2^3\)
Because ((\(2^2\)+\(2^2\)) = 2 *\(2^2\)= \(2^3\))
Similarly
\(2^3\)+ \(2^3\)= \(2^4\)
So accordingly,
\(2^{30}\)= \(2^{29}\)+\(2^{29}\)
Comparing it with the given equation,
X = 29
Y = 29
The value of x+ y = 29 + 29 = 58
Therefore the correct answer will be option D.

Approach Solution 3:
2^x+2^y= 2^30
x+y=?x+y=?
2^x(1+2^y−x)= 2^30
1+2^y−x= 2^30−x
1= 2^30−x−2^y−x

Difference between any two powers of 2 yield 1 if one of the powers is one and the other is zero
i.e. 2^1−2^0= 1
--> 30−x=130−x=1 and y−x=0y−x=0
x= 29x= 29 and x=yx=y
x+y= 58

“For integers x and y, 2^x+2^y=2^30. What is the value of x+y?”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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