$a^{x} = b^{\frac{1}{2}}, b^{y} = c^{\frac{1}{3}}, c^{z} = a^{\frac{1}{2}}$ $a^x = b^(1/2), b^y = c^(1/3), c^z = a^(1/2)$ . What is the value of $x y z$ $xyz$ ?

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$a^{x} = b^{\frac{1}{2}}, b^{y} = c^{\frac{1}{3}}, c^{z} = a^{\frac{1}{2}}$ $a^x = b^(1/2), b^y = c^(1/3), c^z = a^(1/2)$ . What is the value of $x y z$ $xyz$ ?

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Answer 1 · 2016-11-25T15:08:48

$\frac{1}{12}$ $1/12$

Explanation:

Applying $log$ $log$ to the equations

$x log a = \frac{1}{2} log b$ $xloga=1/2logb$
$y log b = \frac{1}{3} log c$ $ylogb=1/3logc$
$z log c = \frac{1}{2} log a$ $zlogc=1/2loga$

then

$x y z = \frac{1}{12} \frac{log b log c log a}{log a log b log c} = \frac{1}{12}$ $xyz=1/12( logb logc log a)/(loga logb logc)=1/12$

Answer 2 · 2016-11-26T08:36:32

$a^{x} = b^{\frac{1}{2}}$ $a^x=b^(1/2)$

$\Rightarrow a^{x y} = {(b^{y})}^{\frac{1}{2}} = {(c^{\frac{1}{3}})}^{\frac{1}{2}} = c^{\frac{1}{6}}$ $=>a^(xy)=(b^y)^(1/2)=(c^(1/3))^(1/2)=c^(1/6)$

$\Rightarrow {(a^{x y})}^{z} = {(c^{z})}^{\frac{1}{6}} = {(a^{\frac{1}{2}})}^{\frac{1}{6}} = a^{\frac{1}{12}}$ $=>(a^(xy))^z=(c^z)^(1/6)=(a^(1/2))^(1/6)=a^(1/12)$

$\Rightarrow a^{x y z} = a^{\frac{1}{12}}$ $=>a^(xyz)=a^(1/12)$

$\Rightarrow x y z = \frac{1}{12}$ $=>xyz=1/12$

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$a^{x} = b^{\frac{1}{2}}, b^{y} = c^{\frac{1}{3}}, c^{z} = a^{\frac{1}{2}}$ $a^x = b^(1/2), b^y = c^(1/3), c^z = a^(1/2)$ . What is the value of $x y z$ $xyz$ ?

2 Answers

Explanation:

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ax=b12,by=c13,cz=a12a^x = b^(1/2), b^y = c^(1/3), c^z = a^(1/2). What is the value of xyzxyz?

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$a^{x} = b^{\frac{1}{2}}, b^{y} = c^{\frac{1}{3}}, c^{z} = a^{\frac{1}{2}}$ $a^x = b^(1/2), b^y = c^(1/3), c^z = a^(1/2)$ . What is the value of $x y z$ $xyz$ ?