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

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

Evaluate

$$24\sqrt{6}+84\approx 142.787753827$$

Solution Steps

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}.

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

Multiply 3 and 2 to get 6.

$$\left(2\sqrt{3}+6\sqrt{2}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2\sqrt{3}+6\sqrt{2}\right)^{2}.

$$4\left(\sqrt{3}\right)^{2}+24\sqrt{3}\sqrt{2}+36\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{3} is 3.

$$4\times 3+24\sqrt{3}\sqrt{2}+36\left(\sqrt{2}\right)^{2}$$

Multiply 4 and 3 to get 12.

$$12+24\sqrt{3}\sqrt{2}+36\left(\sqrt{2}\right)^{2}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$12+24\sqrt{6}+36\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$12+24\sqrt{6}+36\times 2$$

Multiply 36 and 2 to get 72.

$$12+24\sqrt{6}+72$$

Add 12 and 72 to get 84.

$$84+24\sqrt{6}$$

Expand

$$24 \sqrt{6} + 84 = 142.787753827$$

Solution Steps

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}.

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

Multiply 3 and 2 to get 6.

$$\left(2\sqrt{3}+6\sqrt{2}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2\sqrt{3}+6\sqrt{2}\right)^{2}.

$$4\left(\sqrt{3}\right)^{2}+24\sqrt{3}\sqrt{2}+36\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{3} is 3.

$$4\times 3+24\sqrt{3}\sqrt{2}+36\left(\sqrt{2}\right)^{2}$$

Multiply 4 and 3 to get 12.

$$12+24\sqrt{3}\sqrt{2}+36\left(\sqrt{2}\right)^{2}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$12+24\sqrt{6}+36\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$12+24\sqrt{6}+36\times 2$$

Multiply 36 and 2 to get 72.

$$12+24\sqrt{6}+72$$

Add 12 and 72 to get 84.

$$84+24\sqrt{6}$$