11) (8a ^ 2 + b ^ 2)(64a ^ 4 - 8a ^ 2 * b ^ 2 + b ^ 4) 12) (2c ^ 3 - 3p ^ 2)(4c ^ 6 + 6c ^ 3 * p ^ 4 + 9p ^ 4) 13) (4p ^ 4 + 3q ^ 3)(16p ^ 8 - 12p ^ 4 * q ^ 3 + 27q ^ 6) 14) (5x ^…

To simplify each of the given expressions, we will use the identity \( (A + B)(A^2 - AB + B^2) = A^3 + B^3 \), where \( A \) and \( B \) are polynomial expressions.

11) \((8a^2 + b^2)(64a^4 - 8a^2b^2 + b^4)\)

Let \( A = 8a^2 \) and \( B = b^2 \). Hence, we can express the product as follows:

• \( A^3 = (8a^2)^3 = 512a^6 \)
• \( B^3 = (b^2)^3 = b^6 \)

Thus, the expression simplifies to: \[ 512a^6 + b^6 \]

12) \((2c^3 - 3p^2)(4c^6 + 6c^3p^4 + 9p^4)\)

Let \( A = 2c^3 \) and \( B = -3p^2 \). The product simplifies as:

• \( A^3 = (2c^3)^3 = 8c^9 \)
• \( B^3 = (-3p^2)^3 = -27p^6 \)

Thus, the expression simplifies to: \[ 8c^9 - 27p^6 \]

13) \((4p^4 + 3q^3)(16p^8 - 12p^4q^3 + 27q^6)\)

Let \( A = 4p^4 \) and \( B = 3q^3 \). The product simplifies as:

• \( A^3 = (4p^4)^3 = 64p^{12} \)
• \( B^3 = (3q^3)^3 = 27q^9 \)

Thus, the expression simplifies to: \[ 64p^{12} + 27q^9 \]

14) \((5x^2 - 6m^3)(25x^4 + 30x^2m^3 + 36m^6)\)

Let \( A = 5x^2 \) and \( B = -6m^3 \). The product simplifies as:

• \( A^3 = (5x^2)^3 = 125x^6 \)
• \( B^3 = (-6m^3)^3 = -216m^9 \)

Thus, the expression simplifies to: \[ 125x^6 - 216m^9 \]

15) \((7d^5 + 1)(49d^{10} - 7d^5 + 1)\)

Let \( A = 7d^5 \) and \( B = 1 \). The product simplifies as:

• \( A^3 = (7d^5)^3 = 343d^{15} \)
• \( B^3 = 1^3 = 1 \)

Thus, the expression simplifies to: \[ 343d^{15} + 1 \]

Summary of Results:

• 11. \( 512a^6 + b^6 \)
• 12. \( 8c^9 - 27p^6 \)
• 13. \( 64p^{12} + 27q^9 \)
• 14. \( 125x^6 - 216m^9 \)
• 15. \( 343d^{15} + 1 \)