Solve frac{2sqrt{29}+3sqrt{72}}{5sqrt{75}-3sqrt{48}}

$$\frac{ 2 \sqrt{ 29 } +3 \sqrt{ 72 } }{ 5 \sqrt{ 75 } -3 \sqrt{ 48 } }$$

Evaluate

$$\frac{2\sqrt{87}}{39}+\frac{6\sqrt{6}}{13}\approx 1.608860858$$

Solution Steps

Factor 72=6^{2}\times 2. Rewrite the square root of the product \sqrt{6^{2}\times 2} as the product of square roots \sqrt{6^{2}}\sqrt{2}. Take the square root of 6^{2}.

$$\frac{2\sqrt{29}+3\times 6\sqrt{2}}{5\sqrt{75}-3\sqrt{48}}$$

Multiply 3 and 6 to get 18.

$$\frac{2\sqrt{29}+18\sqrt{2}}{5\sqrt{75}-3\sqrt{48}}$$

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$\frac{2\sqrt{29}+18\sqrt{2}}{5\times 5\sqrt{3}-3\sqrt{48}}$$

Multiply 5 and 5 to get 25.

$$\frac{2\sqrt{29}+18\sqrt{2}}{25\sqrt{3}-3\sqrt{48}}$$

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$\frac{2\sqrt{29}+18\sqrt{2}}{25\sqrt{3}-3\times 4\sqrt{3}}$$

Multiply -3 and 4 to get -12.

$$\frac{2\sqrt{29}+18\sqrt{2}}{25\sqrt{3}-12\sqrt{3}}$$

Combine 25\sqrt{3} and -12\sqrt{3} to get 13\sqrt{3}.

$$\frac{2\sqrt{29}+18\sqrt{2}}{13\sqrt{3}}$$

Rationalize the denominator of \frac{2\sqrt{29}+18\sqrt{2}}{13\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\left(2\sqrt{29}+18\sqrt{2}\right)\sqrt{3}}{13\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(2\sqrt{29}+18\sqrt{2}\right)\sqrt{3}}{13\times 3}$$

Multiply 13 and 3 to get 39.

$$\frac{\left(2\sqrt{29}+18\sqrt{2}\right)\sqrt{3}}{39}$$

Use the distributive property to multiply 2\sqrt{29}+18\sqrt{2} by \sqrt{3}.

$$\frac{2\sqrt{29}\sqrt{3}+18\sqrt{2}\sqrt{3}}{39}$$

To multiply \sqrt{29} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{2\sqrt{87}+18\sqrt{2}\sqrt{3}}{39}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{2\sqrt{87}+18\sqrt{6}}{39}$$