Final answer:
The HCF of 8ᵐ³+ⁿ³ and 16ᵐ⁴+m^2ⁿ²+ⁿ⁴ is (2ᵐ+ⁿ)(4ᵐ²-2ᵐⁿ+ⁿ²).
Step-by-step explanation:
To find the highest common factor (HCF) of the given expressions, we need to factorize each expression and find the common factors.
Step 1: Factorize the first expression 8ᵐ³+ⁿ³ into (2ᵐ+ⁿ)(4ᵐ²-2ᵐⁿ+ⁿ²).
Step 2: Factorize the second expression 16ᵐ⁴+m^2ⁿ²+ⁿ⁴ into (2ᵐ²+mⁿ+ⁿ²)(8ᵐ⁴-4ᵐ²ⁿ²+ⁿ⁴).
Step 3: Find the common factors of the two factorizations. The common factors are (2ᵐ+ⁿ) and (4ᵐ²-2ᵐⁿ+ⁿ²). Therefore, the HCF of 8ᵐ³+ⁿ³ and 16ᵐ⁴+m^2ⁿ²+ⁿ⁴ is (2ᵐ+ⁿ)(4ᵐ²-2ᵐⁿ+ⁿ²).