Solve (sqrt{80}+sqrt{90})divsqrt{5}

$$( \sqrt { 80 } + \sqrt { 90 } ) \div \sqrt { 5 }$$

Evaluate

$$3\sqrt{2}+4\approx 8.242640687$$

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}.

$$\frac{4\sqrt{5}+\sqrt{90}}{\sqrt{5}}$$

Factor 90=3^{2}\times 10. Rewrite the square root of the product \sqrt{3^{2}\times 10} as the product of square roots \sqrt{3^{2}}\sqrt{10}. Take the square root of 3^{2}.

$$\frac{4\sqrt{5}+3\sqrt{10}}{\sqrt{5}}$$

Rationalize the denominator of \frac{4\sqrt{5}+3\sqrt{10}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\left(4\sqrt{5}+3\sqrt{10}\right)\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(4\sqrt{5}+3\sqrt{10}\right)\sqrt{5}}{5}$$

Use the distributive property to multiply 4\sqrt{5}+3\sqrt{10} by \sqrt{5}.

$$\frac{4\left(\sqrt{5}\right)^{2}+3\sqrt{10}\sqrt{5}}{5}$$

The square of \sqrt{5} is 5.

$$\frac{4\times 5+3\sqrt{10}\sqrt{5}}{5}$$

Multiply 4 and 5 to get 20.

$$\frac{20+3\sqrt{10}\sqrt{5}}{5}$$

Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.

$$\frac{20+3\sqrt{5}\sqrt{2}\sqrt{5}}{5}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{20+3\times 5\sqrt{2}}{5}$$

Multiply 3 and 5 to get 15.

$$\frac{20+15\sqrt{2}}{5}$$

Divide each term of 20+15\sqrt{2} by 5 to get 4+3\sqrt{2}.

$$4+3\sqrt{2}$$