To simplify the given expression:
[tex]\[
\frac{64^{\frac{-1}{6}} \times 216^{\frac{-1}{3}} \times 81^{\frac{1}{4}}}{512^{\frac{-1}{2}} \times 16^{\frac{1}{4}} \times 9^{\frac{-1}{2}}}
\][/tex]
we will break down and calculate each part separately.
### Step 1: Simplify the Numerator
First, evaluate each component in the numerator:
1. [tex]\( 64^{\frac{-1}{6}} \)[/tex] evaluates to [tex]\( 0.5 \)[/tex].
2. [tex]\( 216^{\frac{-1}{3}} \)[/tex] evaluates to [tex]\( 0.16666666666666669 \)[/tex].
3. [tex]\( 81^{\frac{1}{4}} \)[/tex] evaluates to [tex]\( 3.0 \)[/tex].
Now, multiply these results together:
[tex]\[
64^{\frac{-1}{6}} \times 216^{\frac{-1}{3}} \times 81^{\frac{1}{4}} = 0.5 \times 0.16666666666666669 \times 3.0
\][/tex]
Multiplying these values:
[tex]\[
0.5 \times 0.16666666666666669 \times 3.0 = 0.25
\][/tex]
So the simplified numerator is [tex]\( 0.25 \)[/tex].
### Step 2: Simplify the Denominator
Next, evaluate each component in the denominator:
1. [tex]\( 512^{\frac{-1}{2}} \)[/tex] evaluates to [tex]\( 0.04419417382415922 \)[/tex].
2. [tex]\( 16^{\frac{1}{4}} \)[/tex] evaluates to [tex]\( 2.0 \)[/tex].
3. [tex]\( 9^{\frac{-1}{2}} \)[/tex] evaluates to [tex]\( 0.3333333333333333 \)[/tex].
Now, multiply these results together:
[tex]\[
512^{\frac{-1}{2}} \times 16^{\frac{1}{4}} \times 9^{\frac{-1}{2}} = 0.04419417382415922 \times 2.0 \times 0.3333333333333333
\][/tex]
Multiplying these values:
[tex]\[
0.04419417382415922 \times 2.0 \times 0.3333333333333333 = 0.02946278254943948
\][/tex]
So the simplified denominator is [tex]\( 0.02946278254943948 \)[/tex].
### Step 3: Simplify the Entire Expression
To find the final result, divide the simplified numerator by the simplified denominator:
[tex]\[
\frac{0.25}{0.02946278254943948}
\][/tex]
Dividing these values:
[tex]\[
\frac{0.25}{0.02946278254943948} = 8.48528137423857
\][/tex]
Therefore, the simplified expression is:
[tex]\[
8.48528137423857
\][/tex]
