ax=b12, by=c13, cz=a12 What is the value of xyz?

Step 1 Applying log⁡ to the equations xlog⁡a=12log⁡b ylog⁡b=13log⁡c zlog⁡c=12log⁡a then xyz=112log⁡blog⁡clog⁡alog⁡alog⁡blog⁡c=112

Step 1 ax=b12 ⇒axy=(by)12=(c13)12=c16 ⇒axyz=a112 ⇒xyz=112

Step 1 Applying log⁡ to the equations xlog⁡a=12log⁡b ylog⁡b=13log⁡c zlog⁡c=12log⁡a then xyz=112log⁡blog⁡clog⁡alog⁡alog⁡blog⁡c=112

Step 1 ax=b12 ⇒axy=(by)12=(c13)12=c16 ⇒axyz=a112 ⇒xyz=112

Step by step solution to "ax=b12, by=c13, cz=a12 What is the value of"

High School
Algebra
Pre-Algebra
Equations, expressions, and inequalitie

Cassidy Johnson

Answered question

2022-02-05

a^{x} = b^{\frac{1}{2}}, b^{y} = c^{\frac{1}{3}}, c^{z} = a^{\frac{1}{2}}

What is the value of

x y z ?

Answer & Explanation

Belen Hebert

Beginner2022-02-06Added 8 answers

Step 1
Applying

\log

to the equations

x \log a = \frac{1}{2} \log b

y \log b = \frac{1}{3} \log c

z \log c = \frac{1}{2} \log a

then

x y z = \frac{1}{12} \frac{\log b \log c \log a}{\log a \log b \log c} = \frac{1}{12}

Joe Walker

Beginner2022-02-07Added 9 answers

Step 1
$a^{x} = b^{\frac{1}{2}}$
$\Rightarrow a^{x y} = (b^{y})^{\frac{1}{2}} = (c^{\frac{1}{3}})^{\frac{1}{2}} = c^{\frac{1}{6}}$
$\Rightarrow a^{x y z} = a^{\frac{1}{12}}$
$\Rightarrow x y z = \frac{1}{12}$

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