Solve sqrt{80}-sqrt{20}+(4+sqrt{7})*(4-sqrt{7})

$$\sqrt{ 80 } - \sqrt{ 20 } +(4+ \sqrt{ 7 } ) \times (4- \sqrt{ 7 } )$$

Evaluate

$$2\sqrt{5}+9\approx 13.472135955$$

Solution Steps

Factor 80=4^{2}\times 5. Rewrite the square root of the product \sqrt{4^{2}\times 5} as the product of square roots \sqrt{4^{2}}\sqrt{5}. Take the square root of 4^{2}.

$$4\sqrt{5}-\sqrt{20}+\left(4+\sqrt{7}\right)\left(4-\sqrt{7}\right)$$

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$4\sqrt{5}-2\sqrt{5}+\left(4+\sqrt{7}\right)\left(4-\sqrt{7}\right)$$

Combine 4\sqrt{5} and -2\sqrt{5} to get 2\sqrt{5}.

$$2\sqrt{5}+\left(4+\sqrt{7}\right)\left(4-\sqrt{7}\right)$$

Consider \left(4+\sqrt{7}\right)\left(4-\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$2\sqrt{5}+4^{2}-\left(\sqrt{7}\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$2\sqrt{5}+16-\left(\sqrt{7}\right)^{2}$$

The square of \sqrt{7} is 7.

$$2\sqrt{5}+16-7$$

Subtract 7 from 16 to get 9.

$$2\sqrt{5}+9$$