Question

if 4x=9y=144z show that xy=z(x+2y)​

Answer 1

Given data : $4^{x} = 9^{y} = 144^{z}$

To prove : xy = z(x+2y)

Solution :

$Let\\\\4^{x} = 9^{y} = 144^{z} = k\\\\Such that, \\\\4^{x} = k => 4 = k^{1/x}\\9^{y} = k => 9 = k^{1/y}\\144^{z} = k => 144 = k^{1/z}$

$144=k^{1/z} can -be-written-as :\\\\4^{2} X 9 = k^{1/z}\\\\Replace-4-with - k^{1/x}- and -9 -with - k^{1/y}-as-in :\\\\k^{(1/x)2} X k^{1/y} = k^{1/z}\\\\k^{2/x} X k^{1/y} = k^{1/z}$

We know that aⁿxaᵇ = a⁽ⁿ⁺ᵇ⁾. So,

$k^{2/x} X k^{1/y} = k^{1/z} becomes\\\\k^{(2/x)+(1/y)} = k^{1/z}$

As bases are equal, powers should be equated, i.e.,

$\frac{2}{x} + \frac{1}{y} = \frac{1}{z} \\\\On LCM\\\\\frac{(2y+x)}{xy} = \frac{1}{z} \\\\On-cross-multiplication-and-settling, we-get :\\\\xy = z(x+2y)$

Therefore, if 4x=9y=144z, then xy=z(x+2y). Hence, it is proved.

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