1. **Problem:** Find the highest common factor (HCF) of $12x^{12}$ and $16x^{16}$. 2. **Formula and rules:** - The HCF of two terms is the product of the HCF of their coefficients and the lowest powers of common variables. - For coefficients, find the greatest common divisor (GCD). - For variables with exponents, take the variable with the smallest exponent. 3. **Step-by-step solution:** - Coefficients: $12$ and $16$. The factors of $12$ are $1, 2, 3, 4, 6, 12$. The factors of $16$ are $1, 2, 4, 8, 16$. The greatest common factor is $4$. - Variables: $x^{12}$ and $x^{16}$. The smallest power is $x^{12}$. 4. **Combine results:** $$\text{HCF} = 4x^{12}$$ 5. **Final answer:** The highest common factor of $12x^{12}$ and $16x^{16}$ is $4x^{12}$.