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

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

Evaluate

$$60\sqrt{2}+66\sqrt{5}\approx 232.433300257$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{2}+2\sqrt{5}\right)^{2}.

$$3\sqrt{5}\left(\left(\sqrt{2}\right)^{2}+4\sqrt{2}\sqrt{5}+4\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$3\sqrt{5}\left(2+4\sqrt{2}\sqrt{5}+4\left(\sqrt{5}\right)^{2}\right)$$

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

$$3\sqrt{5}\left(2+4\sqrt{10}+4\left(\sqrt{5}\right)^{2}\right)$$

The square of \sqrt{5} is 5.

$$3\sqrt{5}\left(2+4\sqrt{10}+4\times 5\right)$$

Multiply 4 and 5 to get 20.

$$3\sqrt{5}\left(2+4\sqrt{10}+20\right)$$

Add 2 and 20 to get 22.

$$3\sqrt{5}\left(22+4\sqrt{10}\right)$$

Use the distributive property to multiply 3\sqrt{5} by 22+4\sqrt{10}.

$$66\sqrt{5}+12\sqrt{5}\sqrt{10}$$

Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.

$$66\sqrt{5}+12\sqrt{5}\sqrt{5}\sqrt{2}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$66\sqrt{5}+12\times 5\sqrt{2}$$

Multiply 12 and 5 to get 60.

$$66\sqrt{5}+60\sqrt{2}$$