Solve sqrt{(3sqrt{15})^2+(3sqrt{5})^2}

$$\sqrt { ( 3 \sqrt { 15 } ) ^ { 2 } + ( 3 \sqrt { 5 } ) ^ { 2 } }$$

Evaluate

$$6\sqrt{5}\approx 13.416407865$$

Solution Steps

Expand \left(3\sqrt{15}\right)^{2}.

$$\sqrt{3^{2}\left(\sqrt{15}\right)^{2}+\left(3\sqrt{5}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\sqrt{9\left(\sqrt{15}\right)^{2}+\left(3\sqrt{5}\right)^{2}}$$

The square of \sqrt{15} is 15.

$$\sqrt{9\times 15+\left(3\sqrt{5}\right)^{2}}$$

Multiply 9 and 15 to get 135.

$$\sqrt{135+\left(3\sqrt{5}\right)^{2}}$$

Expand \left(3\sqrt{5}\right)^{2}.

$$\sqrt{135+3^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\sqrt{135+9\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\sqrt{135+9\times 5}$$

Multiply 9 and 5 to get 45.

$$\sqrt{135+45}$$

Add 135 and 45 to get 180.

$$\sqrt{180}$$

Factor 180=6^{2}\times 5. Rewrite the square root of the product \sqrt{6^{2}\times 5} as the product of square roots \sqrt{6^{2}}\sqrt{5}. Take the square root of 6^{2}.

$$6\sqrt{5}$$