Solve frac{sqrt{45}}{sqrt{45}-6}

$$\frac { \sqrt { 45 } } { \sqrt { 45 } - 6 }$$

Evaluate

$$2\sqrt{5}+5\approx 9.472135955$$

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}.

$$\frac{3\sqrt{5}}{\sqrt{45}-6}$$

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}.

$$\frac{3\sqrt{5}}{3\sqrt{5}-6}$$

Rationalize the denominator of \frac{3\sqrt{5}}{3\sqrt{5}-6} by multiplying numerator and denominator by 3\sqrt{5}+6.

$$\frac{3\sqrt{5}\left(3\sqrt{5}+6\right)}{\left(3\sqrt{5}-6\right)\left(3\sqrt{5}+6\right)}$$

Consider \left(3\sqrt{5}-6\right)\left(3\sqrt{5}+6\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{3\sqrt{5}\left(3\sqrt{5}+6\right)}{\left(3\sqrt{5}\right)^{2}-6^{2}}$$

Expand \left(3\sqrt{5}\right)^{2}.

$$\frac{3\sqrt{5}\left(3\sqrt{5}+6\right)}{3^{2}\left(\sqrt{5}\right)^{2}-6^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{3\sqrt{5}\left(3\sqrt{5}+6\right)}{9\left(\sqrt{5}\right)^{2}-6^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{3\sqrt{5}\left(3\sqrt{5}+6\right)}{9\times 5-6^{2}}$$

Multiply 9 and 5 to get 45.

$$\frac{3\sqrt{5}\left(3\sqrt{5}+6\right)}{45-6^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{3\sqrt{5}\left(3\sqrt{5}+6\right)}{45-36}$$

Subtract 36 from 45 to get 9.

$$\frac{3\sqrt{5}\left(3\sqrt{5}+6\right)}{9}$$

Divide 3\sqrt{5}\left(3\sqrt{5}+6\right) by 9 to get \frac{1}{3}\sqrt{5}\left(3\sqrt{5}+6\right).

$$\frac{1}{3}\sqrt{5}\left(3\sqrt{5}+6\right)$$

Use the distributive property to multiply \frac{1}{3}\sqrt{5} by 3\sqrt{5}+6.

$$\frac{1}{3}\sqrt{5}\times 3\sqrt{5}+\frac{1}{3}\sqrt{5}\times 6$$

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

$$\frac{1}{3}\times 5\times 3+\frac{1}{3}\sqrt{5}\times 6$$

Multiply \frac{1}{3} and 5 to get \frac{5}{3}.

$$\frac{5}{3}\times 3+\frac{1}{3}\sqrt{5}\times 6$$

Cancel out 3 and 3.

$$5+\frac{1}{3}\sqrt{5}\times 6$$

Multiply \frac{1}{3} and 6 to get \frac{6}{3}.

$$5+\frac{6}{3}\sqrt{5}$$

Divide 6 by 3 to get 2.

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