To simplify the given expression [tex]\(\frac{2(f^4)^2 f^2}{8 f^{12}}\)[/tex], follow these steps:
1. Simplify the Numerator:
- Inside the numerator, we have [tex]\((f^4)^2\)[/tex].
- Using the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex], we get [tex]\((f^4)^2 = f^{4 \cdot 2} = f^8\)[/tex].
- Then the numerator becomes [tex]\(2 \cdot f^8 \cdot f^2\)[/tex].
2. Combine Exponents in the Numerator:
- Combine the exponents of [tex]\(f\)[/tex] in the numerator using the property [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex].
- Thus [tex]\(2 \cdot f^8 \cdot f^2 = 2 \cdot f^{8+2} = 2f^{10}\)[/tex].
3. Rewrite the Expression:
- The expression now is [tex]\(\frac{2f^{10}}{8f^{12}}\)[/tex].
4. Simplify the Coefficients:
- Simplify the coefficients [tex]\(\frac{2}{8}\)[/tex].
- [tex]\(\frac{2}{8} = \frac{1}{4}\)[/tex].
- So the expression becomes [tex]\(\frac{1 \cdot f^{10}}{4 \cdot f^{12}}\)[/tex].
5. Combine Exponents in the Denominator:
- Using the property for dividing exponents [tex]\(a^m / a^n = a^{m-n}\)[/tex].
- Since [tex]\(f^{10-12} = f^{-2}\)[/tex], the expression becomes [tex]\(\frac{1}{4} \cdot f^{-2}\)[/tex].
6. Rewrite with Positive Exponents:
- Convert the negative exponent into a positive exponent by moving [tex]\(f^{-2}\)[/tex] to the denominator.
- So, [tex]\(\frac{1}{4} f^{-2} = \frac{1}{4f^2}\)[/tex].
Thus, the simplified expression with positive exponents is [tex]\(\boxed{\frac{1}{4f^2}}\)[/tex].
