To simplify the expression \((10a^2b^4)(-2a^{-2}b^6)^2\), we will follow these steps:
- Simplify the expression inside the parentheses \((-2a^{-2}b^6)^2\).
- Multiply the results from step 1 with \(10a^2b^4\).
Step 1: Simplify \((-2a^{-2}b^6)^2\)
\[ (-2a^{-2}b^6)^2 = (-2)^2(a^{-2})^2(b^6)^2 \] \[ = 4a^{-4}b^{12} \]
Step 2: Multiply \(10a^2b^4\) with the result from step 1
Now we can multiply \(10a^{2}b^{4}\) with \(4a^{-4}b^{12}\):
\[ (10a^2b^4)(4a^{-4}b^{12}) = 10 \cdot 4 \cdot a^2 \cdot a^{-4} \cdot b^4 \cdot b^{12} \] \[ = 40 \cdot a^{2 + (-4)} \cdot b^{4 + 12} \] \[ = 40 \cdot a^{-2} \cdot b^{16} \]
Final Result
The simplified expression is:
\[ \frac{40b^{16}}{a^2} \]
Or, if you prefer to keep it in the same format:
\[ 40a^{-2}b^{16} \]