Solve {[17-(sqrt{5}-2sqrt{3})^2]:sqrt{3}-1}^2

$$\{ [ 17 - ( \sqrt { 5 } - 2 \sqrt { 3 } ) ^ { 2 } ] : \sqrt { 3 } - 1 \} ^ { 2 }$$

Evaluate

$$81-8\sqrt{5}\approx 63.11145618$$

Solution Steps

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

$$\left(\frac{17-\left(\left(\sqrt{5}\right)^{2}-4\sqrt{5}\sqrt{3}+4\left(\sqrt{3}\right)^{2}\right)}{\sqrt{3}}-1\right)^{2}$$

The square of \sqrt{5} is 5.

$$\left(\frac{17-\left(5-4\sqrt{5}\sqrt{3}+4\left(\sqrt{3}\right)^{2}\right)}{\sqrt{3}}-1\right)^{2}$$

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

$$\left(\frac{17-\left(5-4\sqrt{15}+4\left(\sqrt{3}\right)^{2}\right)}{\sqrt{3}}-1\right)^{2}$$

The square of \sqrt{3} is 3.

$$\left(\frac{17-\left(5-4\sqrt{15}+4\times 3\right)}{\sqrt{3}}-1\right)^{2}$$

Multiply 4 and 3 to get 12.

$$\left(\frac{17-\left(5-4\sqrt{15}+12\right)}{\sqrt{3}}-1\right)^{2}$$

Add 5 and 12 to get 17.

$$\left(\frac{17-\left(17-4\sqrt{15}\right)}{\sqrt{3}}-1\right)^{2}$$

To find the opposite of 17-4\sqrt{15}, find the opposite of each term.

$$\left(\frac{17-17+4\sqrt{15}}{\sqrt{3}}-1\right)^{2}$$

Subtract 17 from 17 to get 0.

$$\left(\frac{4\sqrt{15}}{\sqrt{3}}-1\right)^{2}$$

Rationalize the denominator of \frac{4\sqrt{15}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\left(\frac{4\sqrt{15}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-1\right)^{2}$$

The square of \sqrt{3} is 3.

$$\left(\frac{4\sqrt{15}\sqrt{3}}{3}-1\right)^{2}$$

Factor 15=3\times 5. Rewrite the square root of the product \sqrt{3\times 5} as the product of square roots \sqrt{3}\sqrt{5}.

$$\left(\frac{4\sqrt{3}\sqrt{5}\sqrt{3}}{3}-1\right)^{2}$$

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

$$\left(\frac{4\times 3\sqrt{5}}{3}-1\right)^{2}$$

Cancel out 3 and 3.

$$\left(4\sqrt{5}-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4\sqrt{5}-1\right)^{2}.

$$16\left(\sqrt{5}\right)^{2}-8\sqrt{5}+1$$

The square of \sqrt{5} is 5.

$$16\times 5-8\sqrt{5}+1$$

Multiply 16 and 5 to get 80.

$$80-8\sqrt{5}+1$$

Add 80 and 1 to get 81.

$$81-8\sqrt{5}$$

Expand

$$81-8\sqrt{5}$$

Solution Steps

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

$$\left(\frac{17-\left(\left(\sqrt{5}\right)^{2}-4\sqrt{5}\sqrt{3}+4\left(\sqrt{3}\right)^{2}\right)}{\sqrt{3}}-1\right)^{2}$$

The square of \sqrt{5} is 5.

$$\left(\frac{17-\left(5-4\sqrt{5}\sqrt{3}+4\left(\sqrt{3}\right)^{2}\right)}{\sqrt{3}}-1\right)^{2}$$

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

$$\left(\frac{17-\left(5-4\sqrt{15}+4\left(\sqrt{3}\right)^{2}\right)}{\sqrt{3}}-1\right)^{2}$$

The square of \sqrt{3} is 3.

$$\left(\frac{17-\left(5-4\sqrt{15}+4\times 3\right)}{\sqrt{3}}-1\right)^{2}$$

Multiply 4 and 3 to get 12.

$$\left(\frac{17-\left(5-4\sqrt{15}+12\right)}{\sqrt{3}}-1\right)^{2}$$

Add 5 and 12 to get 17.

$$\left(\frac{17-\left(17-4\sqrt{15}\right)}{\sqrt{3}}-1\right)^{2}$$

To find the opposite of 17-4\sqrt{15}, find the opposite of each term.

$$\left(\frac{17-17+4\sqrt{15}}{\sqrt{3}}-1\right)^{2}$$

Subtract 17 from 17 to get 0.

$$\left(\frac{4\sqrt{15}}{\sqrt{3}}-1\right)^{2}$$

Rationalize the denominator of \frac{4\sqrt{15}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\left(\frac{4\sqrt{15}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-1\right)^{2}$$

The square of \sqrt{3} is 3.

$$\left(\frac{4\sqrt{15}\sqrt{3}}{3}-1\right)^{2}$$

Factor 15=3\times 5. Rewrite the square root of the product \sqrt{3\times 5} as the product of square roots \sqrt{3}\sqrt{5}.

$$\left(\frac{4\sqrt{3}\sqrt{5}\sqrt{3}}{3}-1\right)^{2}$$

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

$$\left(\frac{4\times 3\sqrt{5}}{3}-1\right)^{2}$$

Cancel out 3 and 3.

$$\left(4\sqrt{5}-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4\sqrt{5}-1\right)^{2}.

$$16\left(\sqrt{5}\right)^{2}-8\sqrt{5}+1$$

The square of \sqrt{5} is 5.

$$16\times 5-8\sqrt{5}+1$$

Multiply 16 and 5 to get 80.

$$80-8\sqrt{5}+1$$

Add 80 and 1 to get 81.

$$81-8\sqrt{5}$$