Solve 3sqrt{45}+2sqrt{12}+3sqrt{5}

$$3 \sqrt{ 45 } +2 \sqrt{ 12 } +3 \sqrt{ 5 }$$

Evaluate

$$4\sqrt{3}\left(\sqrt{15}+1\right)\approx 33.76101896$$

Solution Steps

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}.

$$3\times 3\sqrt{5}+2\sqrt{12}+3\sqrt{5}$$

Multiply 3 and 3 to get 9.

$$9\sqrt{5}+2\sqrt{12}+3\sqrt{5}$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$9\sqrt{5}+2\times 2\sqrt{3}+3\sqrt{5}$$

Multiply 2 and 2 to get 4.

$$9\sqrt{5}+4\sqrt{3}+3\sqrt{5}$$

Combine 9\sqrt{5} and 3\sqrt{5} to get 12\sqrt{5}.

$$12\sqrt{5}+4\sqrt{3}$$