Solve (2sqrt{5}+3sqrt{6})^2

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

Evaluate

$$12\sqrt{30}+74\approx 139.726706901$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2\sqrt{5}+3\sqrt{6}\right)^{2}.

$$4\left(\sqrt{5}\right)^{2}+12\sqrt{5}\sqrt{6}+9\left(\sqrt{6}\right)^{2}$$

The square of \sqrt{5} is 5.

$$4\times 5+12\sqrt{5}\sqrt{6}+9\left(\sqrt{6}\right)^{2}$$

Multiply 4 and 5 to get 20.

$$20+12\sqrt{5}\sqrt{6}+9\left(\sqrt{6}\right)^{2}$$

To multiply \sqrt{5} and \sqrt{6}, multiply the numbers under the square root.

$$20+12\sqrt{30}+9\left(\sqrt{6}\right)^{2}$$

The square of \sqrt{6} is 6.

$$20+12\sqrt{30}+9\times 6$$

Multiply 9 and 6 to get 54.

$$20+12\sqrt{30}+54$$

Add 20 and 54 to get 74.

$$74+12\sqrt{30}$$

Expand

$$12 \sqrt{30} + 74 = 139.726706901$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2\sqrt{5}+3\sqrt{6}\right)^{2}.

$$4\left(\sqrt{5}\right)^{2}+12\sqrt{5}\sqrt{6}+9\left(\sqrt{6}\right)^{2}$$

The square of \sqrt{5} is 5.

$$4\times 5+12\sqrt{5}\sqrt{6}+9\left(\sqrt{6}\right)^{2}$$

Multiply 4 and 5 to get 20.

$$20+12\sqrt{5}\sqrt{6}+9\left(\sqrt{6}\right)^{2}$$

To multiply \sqrt{5} and \sqrt{6}, multiply the numbers under the square root.

$$20+12\sqrt{30}+9\left(\sqrt{6}\right)^{2}$$

The square of \sqrt{6} is 6.

$$20+12\sqrt{30}+9\times 6$$

Multiply 9 and 6 to get 54.

$$20+12\sqrt{30}+54$$

Add 20 and 54 to get 74.

$$74+12\sqrt{30}$$