Question

Решить уравнение: ((125^(x+2) * 16^(x+3) * 5^(x+2))/600^2)=(600^(x+5)/(2^(x+3) * 3^(3x+1))

Answer 1

$125^{(x+2)} *16 ^{x+3} * \frac{ 5^{x+2} }{600 ^{2} } = \frac{600^{x+5} }{2 ^{x+3} } *3 ^{3x+1}$
$(5^3)^{x}(5^3)^2 *(2^4)^{x}(2^4)^3 * \frac{ 5^{x}*5^2 }{(8*25*3) ^{2} } = \frac{(8*3*25)^{x+5} }{2 ^{x}*2^3 } *3 ^{3x} *3$
$5^{3x}*5^6 *2^{4x}*2^{12} * \frac{ 5^{x}*5^2 }{2^6*5^4*3^{2} } = \frac{2^{3(x+5)}*3^{x+5}*5^{2(x+5)} }{2 ^{x}*2^3 } *3 ^{3x} *3$
$5^{4x}*5^4 *2^{4x}*2^{6} * \frac{ 1 }{3^{2} } = \frac{2^{3x}*2^{15}*3^{x}*3^5*5^{2x}*5^{10} }{2 ^{x}*2^3 } *3 ^{3x} *3$
$5^{4x}*5^4 *2^{4x}*2^{6} } = 2^{2x}*2^{12}*3^{x}*3^5*5^{2x}*5^{10}*3 ^{3x} *3*3^2$
$5^{4x}*5^4 *2^{4x}*2^{6} } = 2^{2x}*2^{12}*3^{4x}*3^8*5^{2x}*5^{10}$
$5^{4x+4}*2^{4x+6} = 2^{2x+12}*3^{4x+8}*5^{2x+10}$
$\frac{5^{4x+4}*2^{4x+6} }{2^{2x+12}*3^{4x+8}*5^{2x+10}} =1$
$\frac{5^{4x+4-(2x+10)}*2^{4x+6-(2x+12)} }{3^{4x+8}} =1$
$\frac{5^{4x+4-2x-10}*2^{4x+6-2x-12} }{3^{4x+8}} =1$
$\frac{5^{2x-6}*2^{2x-6} }{3^{4x+8}} =1$
$\frac{(5*2)^{2x-6} }{3^{2(2x+4)}} =1$
$\frac{10^{2x-6} }{9^{2x+4}} =1$