To simplify the expression:
[tex]\[ E = \sqrt[yz]{\frac{a^y}{a^z}} \times \sqrt[zx]{\frac{a^z}{a^x}} \times \sqrt[xy]{\frac{a^x}{a^y}} \][/tex]
we can follow these steps:
### Step 1: Simplify Each Radical
Each part of the product involves a radical term which we need to simplify individually.
1. First term: [tex]\(\sqrt[yz]{\frac{a^y}{a^z}}\)[/tex]
[tex]\[
\sqrt[yz]{\frac{a^y}{a^z}} = \left(\frac{a^y}{a^z}\right)^{\frac{1}{yz}} = \left(a^{y-z}\right)^{\frac{1}{yz}} = a^{\frac{y-z}{yz}}
\][/tex]
2. Second term: [tex]\(\sqrt[zx]{\frac{a^z}{a^x}}\)[/tex]
[tex]\[
\sqrt[zx]{\frac{a^z}{a^x}} = \left(\frac{a^z}{a^x}\right)^{\frac{1}{zx}} = \left(a^{z-x}\right)^{\frac{1}{zx}} = a^{\frac{z-x}{zx}}
\][/tex]
3. Third term: [tex]\(\sqrt[xy]{\frac{a^x}{a^y}}\)[/tex]
[tex]\[
\sqrt[xy]{\frac{a^x}{a^y}} = \left(\frac{a^x}{a^y}\right)^{\frac{1}{xy}} = \left(a^{x-y}\right)^{\frac{1}{xy}} = a^{\frac{x-y}{xy}}
\][/tex]
### Step 2: Combine the Exponents
Now, multiply these simplified terms together:
[tex]\[
E = a^{\frac{y-z}{yz}} \times a^{\frac{z-x}{zx}} \times a^{\frac{x-y}{xy}}
\][/tex]
Using the property of exponents that [tex]\((a^m) \times (a^n) = a^{m+n}\)[/tex]:
[tex]\[
E = a^{\frac{y-z}{yz} + \frac{z-x}{zx} + \frac{x-y}{xy}}
\][/tex]
### Step 3: Simplify the Combined Exponent
To simplify:
[tex]\[
\frac{y-z}{yz} + \frac{z-x}{zx} + \frac{x-y}{xy}
\][/tex]
Find a common denominator, which, in this case, is [tex]\(xyz\)[/tex]:
[tex]\[
\frac{y-z}{yz} = \frac{(y-z)x}{xyz}
\][/tex]
[tex]\[
\frac{z-x}{zx} = \frac{(z-x)y}{xyz}
\][/tex]
[tex]\[
\frac{x-y}{xy} = \frac{(x-y)z}{xyz}
\][/tex]
Combine these fractions:
[tex]\[
\frac{(y-z)x + (z-x)y + (x-y)z}{xyz}
\][/tex]
Simplify the numerator:
[tex]\[
(yz - zx + zy - xy + zx - yz) = 0
\][/tex]
As all terms cancel out, the numerator becomes zero:
[tex]\[
\frac{0}{xyz} = 0
\][/tex]
Thus, the exponent simplifies to zero:
[tex]\[
E = a^0
\][/tex]
### Step 4: Final Simplification
Since any non-zero number raised to the power of 0 is 1:
[tex]\[
E = 1
\][/tex]
Therefore, the simplified expression is:
[tex]\[
1
\][/tex]
