To simplify the expression [tex]\(\frac{4}{(2187)^{\frac{-3}{7}}} - \frac{5}{(256)^{\frac{-1}{4}}} + \frac{2}{\left[(1331)^2\right]^{\frac{-1}{3}}}\)[/tex], we will simplify each term individually and then combine them.
### Step-by-Step Simplification:
#### 1. Simplifying [tex]\(\frac{4}{(2187)^{\frac{-3}{7}}}\)[/tex]:
[tex]\[ \frac{4}{(2187)^{\frac{-3}{7}}} \][/tex]
By the properties of exponents, [tex]\(a^{-b} = \frac{1}{a^b}\)[/tex]. Therefore:
[tex]\[ (2187)^{\frac{-3}{7}} = \frac{1}{(2187)^{\frac{3}{7}}} \][/tex]
Thus:
[tex]\[ \frac{4}{(2187)^{\frac{-3}{7}}} = 4 \cdot (2187)^{\frac{3}{7}} \][/tex]
The given specific numerical result is:
[tex]\[ 4 \cdot (2187)^{\frac{3}{7}} = 108 \][/tex]
So:
[tex]\[ \frac{4}{(2187)^{\frac{-3}{7}}} = 108 \][/tex]
#### 2. Simplifying [tex]\(\frac{5}{(256)^{\frac{-1}{4}}}\)[/tex]:
[tex]\[ \frac{5}{(256)^{\frac{-1}{4}}} \][/tex]
Similarly, using the same exponent rule [tex]\(a^{-b} = \frac{1}{a^b}\)[/tex]:
[tex]\[ (256)^{\frac{-1}{4}} = \frac{1}{(256)^{\frac{1}{4}}} \][/tex]
Thus:
[tex]\[ \frac{5}{(256)^{\frac{-1}{4}}} = 5 \cdot (256)^{\frac{1}{4}} \][/tex]
The given specific numerical result is:
[tex]\[ 5 \cdot (256)^{\frac{1}{4}} = 20 \][/tex]
So:
[tex]\[ \frac{5}{(256)^{\frac{-1}{4}}} = 20 \][/tex]
#### 3. Simplifying [tex]\(\frac{2}{\left[(1331)^2\right]^{\frac{-1}{3}}}\)[/tex]:
First, consider [tex]\(\left[(1331)^2\right]^{\frac{-1}{3}}\)[/tex]:
[tex]\[ \left[(1331)^2\right]^{\frac{-1}{3}} \][/tex]
Using exponent rules [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[ (1331)^{2 \cdot \left(\frac{-1}{3}\right)} = (1331)^{\frac{-2}{3}} \][/tex]
By the property of exponents again, [tex]\(a^{-b} = \frac{1}{a^b}\)[/tex]:
[tex]\[ (1331)^{\frac{-2}{3}} = \frac{1}{(1331)^{\frac{2}{3}}} \][/tex]
Thus:
[tex]\[ \frac{2}{(1331)^{\frac{-2}{3}}} = 2 \cdot (1331)^{\frac{2}{3}} \][/tex]
The given specific numerical result is:
[tex]\[ 2 \cdot (1331)^{\frac{2}{3}} = 242 \][/tex]
So:
[tex]\[ \frac{2}{\left[(1331)^2\right]^{\frac{-1}{3}}} = 242 \][/tex]
#### 4. Combining the Simplified Terms:
Now we combine the simplified terms:
[tex]\[ 108 - 20 + 242 = 330 \][/tex]
### Final Simplified Expression:
[tex]\[
\boxed{330}
\][/tex]
