初中数学题难题一道~若整数x、y、z满足(9/8)^x×(10/9)^y×(16/15)^z=2,求x、y、z的值.
2019-04-12
初中数学题难题一道~
若整数x、y、z满足(9/8)^x×(10/9)^y×(16/15)^z=2,求x、y、z的值.
优质解答
(9/8)^x = (9^x)/(8^x) = [3^(2x)]/[2^(3x)]
(10/9)^y = (10^y)/(9^y) = [(2^y)*(5^y)]/[3^(2y)]
(16/15)^z = (16^z)/(15^z) = [2^(4z)]/[(3^z)*(5^z)]
(9/8)^x×(10/9)^y×(16/15)^z=2
{[3^(2x)]/[2^(3x)]}*{[(2^y)*(5^y)]/[3^(2y)]}*{[2^(4z)]/[(3^z)*(5^z)]}=2
{[3^(2x)][(2^y)*(5^y)][2^(4z)]}/{[2^(3x)][3^(2y)][(3^z)*(5^z)]}=2
[3^(2x-2y-z)][2^(y+4z-3x)][5^(y-z)]=2
因为结果等于2所以
[3^(2x-2y-z)]=1
[2^(y+4z-3x)]=2
[5^(y-z)]=1
即2x-2y-z=0
y+4z-3x=1
y-z=0
解得
x=3
y=2
z=2
(9/8)^x = (9^x)/(8^x) = [3^(2x)]/[2^(3x)]
(10/9)^y = (10^y)/(9^y) = [(2^y)*(5^y)]/[3^(2y)]
(16/15)^z = (16^z)/(15^z) = [2^(4z)]/[(3^z)*(5^z)]
(9/8)^x×(10/9)^y×(16/15)^z=2
{[3^(2x)]/[2^(3x)]}*{[(2^y)*(5^y)]/[3^(2y)]}*{[2^(4z)]/[(3^z)*(5^z)]}=2
{[3^(2x)][(2^y)*(5^y)][2^(4z)]}/{[2^(3x)][3^(2y)][(3^z)*(5^z)]}=2
[3^(2x-2y-z)][2^(y+4z-3x)][5^(y-z)]=2
因为结果等于2所以
[3^(2x-2y-z)]=1
[2^(y+4z-3x)]=2
[5^(y-z)]=1
即2x-2y-z=0
y+4z-3x=1
y-z=0
解得
x=3
y=2
z=2