Solve 4sqrt{80}+8sqrt{45}-4sqrt{48}+sqrt{30,000}

$$4 \sqrt { 80 } + 8 \sqrt { 45 } - 4 \sqrt { 48 } + \sqrt { 30,000 }$$

Evaluate

$$40\sqrt{5}+84\sqrt{3}\approx 234.934986936$$

Solution Steps

Factor 80=4^{2}\times 5. Rewrite the square root of the product \sqrt{4^{2}\times 5} as the product of square roots \sqrt{4^{2}}\sqrt{5}. Take the square root of 4^{2}.

$$4\times 4\sqrt{5}+8\sqrt{45}-4\sqrt{48}+\sqrt{30000}$$

Multiply 4 and 4 to get 16.

$$16\sqrt{5}+8\sqrt{45}-4\sqrt{48}+\sqrt{30000}$$

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$16\sqrt{5}+8\times 3\sqrt{5}-4\sqrt{48}+\sqrt{30000}$$

Multiply 8 and 3 to get 24.

$$16\sqrt{5}+24\sqrt{5}-4\sqrt{48}+\sqrt{30000}$$

Combine 16\sqrt{5} and 24\sqrt{5} to get 40\sqrt{5}.

$$40\sqrt{5}-4\sqrt{48}+\sqrt{30000}$$

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$40\sqrt{5}-4\times 4\sqrt{3}+\sqrt{30000}$$

Multiply -4 and 4 to get -16.

$$40\sqrt{5}-16\sqrt{3}+\sqrt{30000}$$

Factor 30000=100^{2}\times 3. Rewrite the square root of the product \sqrt{100^{2}\times 3} as the product of square roots \sqrt{100^{2}}\sqrt{3}. Take the square root of 100^{2}.

$$40\sqrt{5}-16\sqrt{3}+100\sqrt{3}$$

Combine -16\sqrt{3} and 100\sqrt{3} to get 84\sqrt{3}.

$$40\sqrt{5}+84\sqrt{3}$$