c) \( \sqrt{1400 a^{6} b^{5}} \)

To simplify \( \sqrt{1400 a^{6} b^{5}} \), start by breaking down the square root into parts: 1. Factor \( 1400 \): \[ 1400 = 14 \times 100 = 14 \times 10^2 = 14 \times (2 \times 5)^2 = 2^2 \times 5^2 \times 7. \] Thus, the prime factorization gives us \( \sqrt{1400} = \sqrt{2^2 \times 5^2 \times 7} = 2 \times 5 \times \sqrt{7} = 10\sqrt{7} \). 2. Now for the variables: \[ \sqrt{a^6} = a^{6/2} = a^3 \quad \text{and} \quad \sqrt{b^5} = \sqrt{b^4 \times b} = \sqrt{b^4} \times \sqrt{b} = b^{4/2} \times b^{1/2} = b^2 \sqrt{b}. \] 3. Combine all parts together: \[ \sqrt{1400 a^6 b^5} = 10 \sqrt{7} \cdot a^3 \cdot b^2 \sqrt{b} = 10 a^3 b^2 \sqrt{7b}. \] So, the simplified expression is: \[ \boxed{10 a^3 b^2 \sqrt{7b}}. \]