If x and y are integers and 72^x∗54^y=96, what is the value of x−y?

Question

If x and y are integers and  72^x∗54^y=96 , what is the value of x−y?

A. -1
B. 0
C. 1
D. 2
E. 3

0akshay0 · Accepted Answer

72^x∗54^y=96 (3^2*2^3)^x∗(3^3*2)^y=2^5*3 3^{2x}*2^{3y}*3^{3y}*2^y=2^5*3 3^{2x+3y}*2^{3x+y}=2^5*3 on solving 2x+3y=1 and 3x+y=5 we get x=2 and y=-1 so x-y = 2-(-1) =3 Hence option E is correct Hit Kudos if you liked it 8-)

Missyy · Answer

72^x∗54^y=96 divide 6 at both side of the equation:
12^x*9^y=16 divide 3 at both side of the equation:
4^x*3^y=16/3
So x=2, y=-1

niks18 · Answer

Hi  Missyy

You cannot divide both sides of the exponents and use the quotients in a manner that you did.

for eg if we have  6^2*9^2  and as per you approach if I divide this by 3, then I should get  2^2*3^2=36

but actually it will be  \frac{6*6*9*9}{3}=972 .

Moreover the division is also wrong.  72^x*54^y  is one number but you are individually dividing both  72^x  &  54^y  by 6 individually. Had there been a + or - then individual division might be possible but in case of variable exponents it is incorrect

KanishkM · Answer

Be careful in such questions

72^x * 54^y = 96

Convert the above into their factors

^x *  ^y = 2^5 * 3^1 
2^3x * 2^y * 3^2x * 3^3y = 2^5 * 3^1

3x + y = 5
2x + 3y = 1

Solve them to get x =2 & y =-1

E

NinetyFour · Answer

My reasoning:

72 broken down into primes is  2^3 * 3^2  After multiplying  x  in we get  {2^3x} * 3^{2x}

54 broken down into primes is  2 * 3^3  After multiplying  y  in we get  2^y * 3^{3y}

96 broken down into primes is  2^5 * 3

Same base means you can add the exponents together so we get:  2^{(3x+y)} * 3^{(2x+3y)}

We know  2^{(3x+y)} = 2^5  and  3^{(2x+3y)} = 3^1

We can solve for x and y now which gives us 2 and -1 respectively. Subtract them and we get 3 (Choice E)

Malar95 · Answer

My brain freezes with these kind of Q's. It was a guess answer but excluding the total, I did notice a constant play of '9's and '6's in the Q. So I felt the best answer could be 3. I may have gotten lucky.

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If x and y are integers and 72^x∗54^y=96, what is the value of x−y?

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If x and y are integers and 72^x∗54^y=96, what is the value of x−y?

Prep Toolkit

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My Rewards