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Multiply. Write your answer in simplest form.\newline66×39\sqrt{66} \times \sqrt{39}

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Full solution

Q. Multiply. Write your answer in simplest form.\newline66×39\sqrt{66} \times \sqrt{39}
  1. Find Prime Factorization: Find the prime factorization of 66\sqrt{66} and 39\sqrt{39}. The prime factorization of 6666 is 2×3×112 \times 3 \times 11, and the prime factorization of 3939 is 3×133 \times 13. So, 66×39\sqrt{66} \times \sqrt{39} can be expressed as 2×3×11×3×13\sqrt{2 \times 3 \times 11} \times \sqrt{3 \times 13}.
  2. Apply Multiplication Property: Apply the multiplication property of square roots: 2×3×11×3×13=2×3×11×3×13\sqrt{2 \times 3 \times 11} \times \sqrt{3 \times 13} = \sqrt{2 \times 3 \times 11 \times 3 \times 13}.
  3. Group Perfect Square Factors: Group the perfect square factors: 2×3×11×3×13=2×32×11×13\sqrt{2 \times 3 \times 11 \times 3 \times 13} = \sqrt{2 \times 3^2 \times 11 \times 13}.
  4. Pull Out Base: Pull out the base of the perfect square factor: 2×32×11×13=3×2×11×13\sqrt{2 \times 3^2 \times 11 \times 13} = 3 \times \sqrt{2 \times 11 \times 13}.
  5. Simplify Expression: Simplify 3×2×11×133 \times \sqrt{2 \times 11 \times 13}: 66×39=3×2×11×13=3×286\sqrt{66} \times \sqrt{39} = 3 \times \sqrt{2 \times 11 \times 13} = 3 \times \sqrt{286}.

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