An algebra problem by Fardin Islam

Algebra Level 1

9^x=(27)^y

If x/y=a/b, Then what is the value of a+b?

5 solutions

Vinayak Srivastava
May 15, 2020

$9^x=27^y$

$3^{2x}=3^{3y}$

Since base is same, exponents are equal.

$2x=3y$

$\implies \dfrac{x}{y} = \dfrac{3}{2} = \dfrac{a}{b}$

Assuming $a,b$ to be co-prime, positive integers,

$\dfrac{a}{b} = \dfrac{3}{2}$

$\implies a=3, b=2$

$\implies a+b=3+2=\boxed{5}$

if a/b=3/2 this does not mean that a=3 and b=2 it is only if HCF(a,b)=1 else you can take

a=3, 6, 9... and b=2, 4, 6...

=> a+b=5, 10, 15, 20...

you can not assume a, b to be co-prime on your will it must also be mentioned in the question

Zakir Husain - 1 year ago

It is not my mistake if it is not mentioned in the question. I also knew that therefore I wrote "Assuming a,b to be co-prime, positive integers".

Vinayak Srivastava - 1 year ago

Sahar Bano
Mar 13, 2020

This can be simplified as

3^(2x) = 3^(3y)

=> 2x=3y

=>x/y=3/2 =>a=3 ; b=2

As 3+2=5

Hence the answer is 5 Note: Their can be more than one answer as it is not given that GCD(a,b)=1

Other answers can be of the form a=3n; b=2n

=>a+b=5n where n is an integer

Rob Funicella
Jul 16, 2015

(3x3)^3=(3x3x3)^2

(3x3)x(3x3)x(3x3) = (3x3x3)x(3x3x3)

This makes sense to me.

Sadasiva Panicker
Jul 12, 2015

9 raised3 = 27 raised2 therefore x/y = a/b =3/2 Then a + b + 3 + 2 = 5

Fardin Islam
May 24, 2015

Given 9^x=27^y Or, 3^2x=3^3y Or,2x=3y Or, x/y=3/2 Then, a+b=3+2=5

The answer can be any multiple of 5 not necessarily 5 only.

But the smallest possible answer is 5.

Sahar Bano - 1 year, 3 months ago

a/b=3/2 is a ratio so how come a+b=5
It should be somewhat like a+b=3x+2x and not simply 5

Chaitnya Shrivastava - 5 years, 11 months ago