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

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

Evaluate

$$12\sqrt{15}+57\approx 103.475800154$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3\sqrt{5}+2\sqrt{3}\right)^{2}.

$$9\left(\sqrt{5}\right)^{2}+12\sqrt{5}\sqrt{3}+4\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{5} is 5.

$$9\times 5+12\sqrt{5}\sqrt{3}+4\left(\sqrt{3}\right)^{2}$$

Multiply 9 and 5 to get 45.

$$45+12\sqrt{5}\sqrt{3}+4\left(\sqrt{3}\right)^{2}$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$45+12\sqrt{15}+4\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$45+12\sqrt{15}+4\times 3$$

Multiply 4 and 3 to get 12.

$$45+12\sqrt{15}+12$$

Add 45 and 12 to get 57.

$$57+12\sqrt{15}$$

Expand

$$12 \sqrt{15} + 57 = 103.475800154$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3\sqrt{5}+2\sqrt{3}\right)^{2}.

$$9\left(\sqrt{5}\right)^{2}+12\sqrt{5}\sqrt{3}+4\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{5} is 5.

$$9\times 5+12\sqrt{5}\sqrt{3}+4\left(\sqrt{3}\right)^{2}$$

Multiply 9 and 5 to get 45.

$$45+12\sqrt{5}\sqrt{3}+4\left(\sqrt{3}\right)^{2}$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$45+12\sqrt{15}+4\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$45+12\sqrt{15}+4\times 3$$

Multiply 4 and 3 to get 12.

$$45+12\sqrt{15}+12$$

Add 45 and 12 to get 57.

$$57+12\sqrt{15}$$