Solve (5sqrt{3}+3sqrt{55})^2

$$( 5 \sqrt { 3 } + 3 \sqrt { 55 } ) ^ { 2 }$$

Evaluate

$$30\left(\sqrt{165}+19\right)\approx 955.35697736$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5\sqrt{3}+3\sqrt{55}\right)^{2}.

$$25\left(\sqrt{3}\right)^{2}+30\sqrt{3}\sqrt{55}+9\left(\sqrt{55}\right)^{2}$$

The square of \sqrt{3} is 3.

$$25\times 3+30\sqrt{3}\sqrt{55}+9\left(\sqrt{55}\right)^{2}$$

Multiply 25 and 3 to get 75.

$$75+30\sqrt{3}\sqrt{55}+9\left(\sqrt{55}\right)^{2}$$

To multiply \sqrt{3} and \sqrt{55}, multiply the numbers under the square root.

$$75+30\sqrt{165}+9\left(\sqrt{55}\right)^{2}$$

The square of \sqrt{55} is 55.

$$75+30\sqrt{165}+9\times 55$$

Multiply 9 and 55 to get 495.

$$75+30\sqrt{165}+495$$

Add 75 and 495 to get 570.

$$570+30\sqrt{165}$$

Expand

$$30 \sqrt{165} + 570 = 955.35697736$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5\sqrt{3}+3\sqrt{55}\right)^{2}.

$$25\left(\sqrt{3}\right)^{2}+30\sqrt{3}\sqrt{55}+9\left(\sqrt{55}\right)^{2}$$

The square of \sqrt{3} is 3.

$$25\times 3+30\sqrt{3}\sqrt{55}+9\left(\sqrt{55}\right)^{2}$$

Multiply 25 and 3 to get 75.

$$75+30\sqrt{3}\sqrt{55}+9\left(\sqrt{55}\right)^{2}$$

To multiply \sqrt{3} and \sqrt{55}, multiply the numbers under the square root.

$$75+30\sqrt{165}+9\left(\sqrt{55}\right)^{2}$$

The square of \sqrt{55} is 55.

$$75+30\sqrt{165}+9\times 55$$

Multiply 9 and 55 to get 495.

$$75+30\sqrt{165}+495$$

Add 75 and 495 to get 570.

$$570+30\sqrt{165}$$