Solve sqrt{5}(3sqrt{5}+sqrt{2})+sqrt{2}(2sqrt{8}+sqrt{5})

$$\sqrt { 5 } ( 3 \sqrt { 5 } + \sqrt { 2 } ) + \sqrt { 2 } ( 2 \sqrt { 8 } + \sqrt { 5 } )$$

Evaluate

$$2\sqrt{10}+23\approx 29.32455532$$

Solution Steps

Use the distributive property to multiply \sqrt{5} by 3\sqrt{5}+\sqrt{2}.

$$3\left(\sqrt{5}\right)^{2}+\sqrt{5}\sqrt{2}+\sqrt{2}\left(2\sqrt{8}+\sqrt{5}\right)$$

The square of \sqrt{5} is 5.

$$3\times 5+\sqrt{5}\sqrt{2}+\sqrt{2}\left(2\sqrt{8}+\sqrt{5}\right)$$

Multiply 3 and 5 to get 15.

$$15+\sqrt{5}\sqrt{2}+\sqrt{2}\left(2\sqrt{8}+\sqrt{5}\right)$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$15+\sqrt{10}+\sqrt{2}\left(2\sqrt{8}+\sqrt{5}\right)$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$15+\sqrt{10}+\sqrt{2}\left(2\times 2\sqrt{2}+\sqrt{5}\right)$$

Multiply 2 and 2 to get 4.

$$15+\sqrt{10}+\sqrt{2}\left(4\sqrt{2}+\sqrt{5}\right)$$

Use the distributive property to multiply \sqrt{2} by 4\sqrt{2}+\sqrt{5}.

$$15+\sqrt{10}+4\left(\sqrt{2}\right)^{2}+\sqrt{2}\sqrt{5}$$

The square of \sqrt{2} is 2.

$$15+\sqrt{10}+4\times 2+\sqrt{2}\sqrt{5}$$

Multiply 4 and 2 to get 8.

$$15+\sqrt{10}+8+\sqrt{2}\sqrt{5}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$15+\sqrt{10}+8+\sqrt{10}$$

Add 15 and 8 to get 23.

$$23+\sqrt{10}+\sqrt{10}$$

Combine \sqrt{10} and \sqrt{10} to get 2\sqrt{10}.

$$23+2\sqrt{10}$$