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

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

Evaluate

$$5\sqrt{3}+7\sqrt{2}\approx 18.559748974$$

Solution Steps

Multiply \sqrt{3}+\sqrt{2} and \sqrt{3}+\sqrt{2} to get \left(\sqrt{3}+\sqrt{2}\right)^{2}.

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

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

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

The square of \sqrt{3} is 3.

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

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

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

The square of \sqrt{2} is 2.

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

Add 3 and 2 to get 5.

$$\left(5+2\sqrt{6}\right)\sqrt{3}+\sqrt{2}$$

Use the distributive property to multiply 5+2\sqrt{6} by \sqrt{3}.

$$5\sqrt{3}+2\sqrt{6}\sqrt{3}+\sqrt{2}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

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

Multiply \sqrt{3} and \sqrt{3} to get 3.

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

Multiply 2 and 3 to get 6.

$$5\sqrt{3}+6\sqrt{2}+\sqrt{2}$$

Combine 6\sqrt{2} and \sqrt{2} to get 7\sqrt{2}.

$$5\sqrt{3}+7\sqrt{2}$$