Solve sqrt{80}+sqrt{288}+sqrt{32}+10sqrt{18}

$$\sqrt { 80 } + \sqrt { 288 } + \sqrt { 32 } + 10 \sqrt { 18 }$$

Evaluate

$$2\sqrt{2}\left(\sqrt{10}+23\right)\approx 73.998095779$$

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}.

$$4\sqrt{5}+\sqrt{288}+\sqrt{32}+10\sqrt{18}$$

Factor 288=12^{2}\times 2. Rewrite the square root of the product \sqrt{12^{2}\times 2} as the product of square roots \sqrt{12^{2}}\sqrt{2}. Take the square root of 12^{2}.

$$4\sqrt{5}+12\sqrt{2}+\sqrt{32}+10\sqrt{18}$$

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

$$4\sqrt{5}+12\sqrt{2}+4\sqrt{2}+10\sqrt{18}$$

Combine 12\sqrt{2} and 4\sqrt{2} to get 16\sqrt{2}.

$$4\sqrt{5}+16\sqrt{2}+10\sqrt{18}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$4\sqrt{5}+16\sqrt{2}+10\times 3\sqrt{2}$$

Multiply 10 and 3 to get 30.

$$4\sqrt{5}+16\sqrt{2}+30\sqrt{2}$$

Combine 16\sqrt{2} and 30\sqrt{2} to get 46\sqrt{2}.

$$4\sqrt{5}+46\sqrt{2}$$