Solve {left(4sqrt{15}-2sqrt{39}right)}^2

$${ \left(4 \sqrt{ 15 } -2 \sqrt{ 39 } \right) }^{ 2 }$$

Evaluate

$$396-48\sqrt{65}\approx 9.011628082$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4\sqrt{15}-2\sqrt{39}\right)^{2}.

$$16\left(\sqrt{15}\right)^{2}-16\sqrt{15}\sqrt{39}+4\left(\sqrt{39}\right)^{2}$$

The square of \sqrt{15} is 15.

$$16\times 15-16\sqrt{15}\sqrt{39}+4\left(\sqrt{39}\right)^{2}$$

Multiply 16 and 15 to get 240.

$$240-16\sqrt{15}\sqrt{39}+4\left(\sqrt{39}\right)^{2}$$

To multiply \sqrt{15} and \sqrt{39}, multiply the numbers under the square root.

$$240-16\sqrt{585}+4\left(\sqrt{39}\right)^{2}$$

The square of \sqrt{39} is 39.

$$240-16\sqrt{585}+4\times 39$$

Multiply 4 and 39 to get 156.

$$240-16\sqrt{585}+156$$

Add 240 and 156 to get 396.

$$396-16\sqrt{585}$$

Factor 585=3^{2}\times 65. Rewrite the square root of the product \sqrt{3^{2}\times 65} as the product of square roots \sqrt{3^{2}}\sqrt{65}. Take the square root of 3^{2}.

$$396-16\times 3\sqrt{65}$$

Multiply -16 and 3 to get -48.

$$396-48\sqrt{65}$$

Expand

$$396-48\sqrt{65}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4\sqrt{15}-2\sqrt{39}\right)^{2}.

$$16\left(\sqrt{15}\right)^{2}-16\sqrt{15}\sqrt{39}+4\left(\sqrt{39}\right)^{2}$$

The square of \sqrt{15} is 15.

$$16\times 15-16\sqrt{15}\sqrt{39}+4\left(\sqrt{39}\right)^{2}$$

Multiply 16 and 15 to get 240.

$$240-16\sqrt{15}\sqrt{39}+4\left(\sqrt{39}\right)^{2}$$

To multiply \sqrt{15} and \sqrt{39}, multiply the numbers under the square root.

$$240-16\sqrt{585}+4\left(\sqrt{39}\right)^{2}$$

The square of \sqrt{39} is 39.

$$240-16\sqrt{585}+4\times 39$$

Multiply 4 and 39 to get 156.

$$240-16\sqrt{585}+156$$

Add 240 and 156 to get 396.

$$396-16\sqrt{585}$$

Factor 585=3^{2}\times 65. Rewrite the square root of the product \sqrt{3^{2}\times 65} as the product of square roots \sqrt{3^{2}}\sqrt{65}. Take the square root of 3^{2}.

$$396-16\times 3\sqrt{65}$$

Multiply -16 and 3 to get -48.

$$396-48\sqrt{65}$$