To solve for [tex]\(\left(5 a^2 b^{-3} c^{-4}\right)^2\)[/tex], let's break down the process step-by-step.
### Step 1: Understand the Expression
We start with the given expression inside the parentheses:
[tex]\[ 5 a^2 b^{-3} c^{-4} \][/tex]
### Step 2: Apply the Exponent
We need to square the entire expression:
[tex]\[ \left( 5 a^2 b^{-3} c^{-4} \right)^2 \][/tex]
### Step 3: Distribute the Exponent to Each Term
When we square a product, we apply the exponent to each factor inside the parentheses individually:
[tex]\[ (5)^2 \cdot (a^2)^2 \cdot (b^{-3})^2 \cdot (c^{-4})^2 \][/tex]
### Step 4: Calculate Each Term
Now, let's calculate each of these terms separately:
1. [tex]\((5)^2 = 25\)[/tex]
2. [tex]\((a^2)^2 = a^{2 \cdot 2} = a^4\)[/tex]
3. [tex]\((b^{-3})^2 = b^{-3 \cdot 2} = b^{-6}\)[/tex]
4. [tex]\((c^{-4})^2 = c^{-4 \cdot 2} = c^{-8}\)[/tex]
### Step 5: Combine the Results
Combine all these results together:
[tex]\[ 25 \cdot a^4 \cdot b^{-6} \cdot c^{-8} \][/tex]
### Step 6: Simplify the Expression
We can write the final simplified expression. Since [tex]\( b^{-6} \)[/tex] and [tex]\( c^{-8} \)[/tex] are negative exponents, we can write them in the denominator to obtain a more standard form:
[tex]\[ \frac{25 a^4}{b^6 c^8} \][/tex]
### Final Answer
The simplified expression is:
[tex]\[ \boxed{\frac{25 a^4}{b^6 c^8}} \][/tex]
