If (2^x)(3^y) = 288, where x and y are positive integers, then (2^x-1)

If (2^x)(3^y) = 288, where x and y are positive integers, then (2^x-1)(3^y-2) equals:

16
24
48
96
144


So I would start attacking this problem by quickly performing the prime factorization of 288. With that it is easy to count the 5 twos and the 2 threes that are the prime factors. So x=5, y=2. now quickly 2^4(3^0)=16. Than answer should be number 1.

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Thanks,
Skip

Thanks, I didn't even think to use the prime factorization.

Algebraically we can approach the problem in another way as well.

If (2^x)(3^y) = 288
then (2^x-1)(3^y-2) = 288/(2[*]3[*]3)
The reason being the exponent of 2 is reduced by 1 and exponent of 3 is reduced by 2.
Hence, the answer would be 288/18 = 16.

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Please give kudos, if my comments were helpful.

Hi Skip, could you please explain prime factorization and how you count the three's. I think I see how you count the two's by taking 2, 8, and 8 seperately and see how many two's making up each number but I don't quite understand how you get the three's.

Thanks, Steven

Here's how I do prime factorization.

In one row you start with your number to be factored and in the other column you write the smallest possible prime that number is divisible by. Then you do the same for the (number/prime) and so on until you get to 1.

For example:

number:______288______144______72______36______18______9______3______1 (done)
prime factor:_______2_________2______ 2_______2_______2______3______3

In this case you it's easy to see that 288 = 2*2*2*2*2*3*3 = 2^5 * 3^2.

I approached it by using prime factorization. you need to know how many two's and three's are there in 288. using prime factorization, i got 4 two's and 2 three's.
you just need to plug in the values for x and y and solve the given equation.
(2^5-1)(3^2-2)= (2^4)(3^0)= (16)(1)= 16

hope that helps

Prime factorize 288 then do your magic:

\(\left( { 2 }^{ x } \right) \left( 3^{ y } \right) =288\\ \left( { 2 }^{ x } \right) \left( 3^{ y } \right) ={ 2 }^{ 5 }{ 3 }^{ 2 }\\ x=5\\ y=2\\ \\ \left( { 2 }^{ x-1 } \right) \left( { 3 }^{ y-2 } \right) =\left( { 2 }^{ 5-1 } \right) \left( { 3 }^{ 2-2 } \right) ={ 2 }^{ 4 }{ 3 }^{ 0 }=16\)
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\(288 = 3 *96 = 3 * 3*32 = 3*3* 4*8 = 3*3 * 2*2 * 2*2*2 = 3^22^5\)

\(2^x*3^y = 288\)

\(2^x3^y = 2^53^2\)

x = 5 and y = 2.

\(2^{x-1}*3^{y-2}\)

=\(2^{5-1} *3^{2-2}\)

=\(2^4 * 3^0\)

=16

E is our answer.

Break down 288 into its prime factors. You find that X = 5 and Y = 2. The rest is...

288= 2*2*2*2*2*3*3 = 2^5 * 3^2

(2^x-1) * (3^y-2) = 2^4 * 3^2-2
= 16 *1= 16
Answer A.

If we break up the term as the following then \(2^{x-1} * 3^{y-2} = \frac{2^x}{2} * \frac{3^y}{3^2}\)

Since \(2^{x} * 3^{y} = 288\), substituting, we get \(\frac{288}{2 * 9} = \frac{32}{2} = 16\)


Option A

Arun Kumar
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APPROACH #1: Use prime factorization
\(288 = (2)(2)(2)(2)(2)(3)(3) = (2^5)(3^2)\)
So, the given equation becomes: \((2^x)(3^y) = (2^5)(3^2)\), which means \(x=5\) and \(y=2\)
Thus \((2^{x-1})(3^{y-2}) = (2^{5-1})(3^{2-2}) = (2^{4})(3^{0}) = (16)(1) = 16\)
Answer: A

APPROACH #2: Use algebraic manipulation
Given: \((2^x)(3^y) = 288\)
Divide both sides of the equation by \(2^1\) (aka \(2\)) to get: \(\frac{(2^x)(3^y)}{2^1} = \frac{288}{2^1}\)
Simplify to get: \((2^{x-1})(3^y) = 144\)
Divide both sides of the equation by \(3^2\) to get: \(\frac{(2^{x-1})(3^y)}{3^2} = \frac{144}{3^2}\)
Simplify: \((2^{x-1})(3^{y-2}) = 16\)
Answer: A
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