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Algebra Level 2

$\large { \begin{aligned}27^x - 27^y &=& 24\\ 9^x + 3^{x+y} + 9^y &=& 6 \\ 3^x - 3^y &=& \ ? \end{aligned}}$

4 solutions

Noel Lo
May 12, 2015

Let $a=3^x$ and $b = 3^y$ .

We have: $a^3 - b^3 = 24$ and $a^2+ ab + b^2 = 6$

Since $a^3 - b^3 = (a^2 + ab + b^2)(a-b)$ , we have $6(a-b) = 24$ so that $a-b = 3^x - 3^y = \frac{24}{6} = \boxed{4}$ .

Did it the same way

Athiyaman Nallathambi - 5 years, 11 months ago

Nice solution

Anil Patel - 5 years, 10 months ago

Hadia Qadir
Jul 22, 2015

let 3 ^x=a and 3^y=b then 27^x-27^y=24 or a^3-b^3=24 or (a-b) (a^2+ab+b^2)=24 or a-b=24/6=4

Tootie Frootie
May 20, 2015

4 ... Replace 3^x as 'a' and 3^y as 'b'. (a-b) x (a^2+ab+b^2) = (a^3 - b^3). So, (a-b) is 24/6 = 4.

Saheli Raha
May 14, 2015

let 3 ^x=a and 3^y=b then 27^x-27^y=24 or a^3-b^3=24 or (a-b) (a^2+ab+b^2)=24 or a-b=24/6=4