Solve 2/5+665sqrt{45}

$$\frac{ 2 }{ 5+665 \sqrt{ 45 } }$$

Evaluate

$$\frac{399\sqrt{5}-1}{1990010}\approx 0.000447832$$

Solution Steps

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$\frac{2}{5+665\times 3\sqrt{5}}$$

Multiply 665 and 3 to get 1995.

$$\frac{2}{5+1995\sqrt{5}}$$

Rationalize the denominator of \frac{2}{5+1995\sqrt{5}} by multiplying numerator and denominator by 5-1995\sqrt{5}.

$$\frac{2\left(5-1995\sqrt{5}\right)}{\left(5+1995\sqrt{5}\right)\left(5-1995\sqrt{5}\right)}$$

Consider \left(5+1995\sqrt{5}\right)\left(5-1995\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\left(5-1995\sqrt{5}\right)}{5^{2}-\left(1995\sqrt{5}\right)^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{2\left(5-1995\sqrt{5}\right)}{25-\left(1995\sqrt{5}\right)^{2}}$$

Expand \left(1995\sqrt{5}\right)^{2}.

$$\frac{2\left(5-1995\sqrt{5}\right)}{25-1995^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate 1995 to the power of 2 and get 3980025.

$$\frac{2\left(5-1995\sqrt{5}\right)}{25-3980025\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{2\left(5-1995\sqrt{5}\right)}{25-3980025\times 5}$$

Multiply 3980025 and 5 to get 19900125.

$$\frac{2\left(5-1995\sqrt{5}\right)}{25-19900125}$$

Subtract 19900125 from 25 to get -19900100.

$$\frac{2\left(5-1995\sqrt{5}\right)}{-19900100}$$

Divide 2\left(5-1995\sqrt{5}\right) by -19900100 to get -\frac{1}{9950050}\left(5-1995\sqrt{5}\right).

$$-\frac{1}{9950050}\left(5-1995\sqrt{5}\right)$$

Use the distributive property to multiply -\frac{1}{9950050} by 5-1995\sqrt{5}.

$$-\frac{1}{9950050}\times 5-\frac{1}{9950050}\left(-1995\right)\sqrt{5}$$

Express -\frac{1}{9950050}\times 5 as a single fraction.

$$\frac{-5}{9950050}-\frac{1}{9950050}\left(-1995\right)\sqrt{5}$$

Reduce the fraction \frac{-5}{9950050} to lowest terms by extracting and canceling out 5.

$$-\frac{1}{1990010}-\frac{1}{9950050}\left(-1995\right)\sqrt{5}$$

Express -\frac{1}{9950050}\left(-1995\right) as a single fraction.

$$-\frac{1}{1990010}+\frac{-\left(-1995\right)}{9950050}\sqrt{5}$$

Multiply -1 and -1995 to get 1995.

$$-\frac{1}{1990010}+\frac{1995}{9950050}\sqrt{5}$$

Reduce the fraction \frac{1995}{9950050} to lowest terms by extracting and canceling out 5.

$$-\frac{1}{1990010}+\frac{399}{1990010}\sqrt{5}$$