Solve 1/4sqrt{80}-1/6sqrt{63}-1/9sqrt{180}

$$\frac{ 1 }{ 4 } \sqrt{ 80 } - \frac{ 1 }{ 6 } \sqrt{ 63 } - \frac{ 1 }{ 9 } \sqrt{ 180 }$$

Evaluate

$$\frac{\sqrt{5}}{3}-\frac{\sqrt{7}}{2}\approx -0.577519663$$

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}.

$$\frac{1}{4}\times 4\sqrt{5}-\frac{1}{6}\sqrt{63}-\frac{1}{9}\sqrt{180}$$

Cancel out 4 and 4.

$$\sqrt{5}-\frac{1}{6}\sqrt{63}-\frac{1}{9}\sqrt{180}$$

Factor 63=3^{2}\times 7. Rewrite the square root of the product \sqrt{3^{2}\times 7} as the product of square roots \sqrt{3^{2}}\sqrt{7}. Take the square root of 3^{2}.

$$\sqrt{5}-\frac{1}{6}\times 3\sqrt{7}-\frac{1}{9}\sqrt{180}$$

Express -\frac{1}{6}\times 3 as a single fraction.

$$\sqrt{5}+\frac{-3}{6}\sqrt{7}-\frac{1}{9}\sqrt{180}$$

Reduce the fraction \frac{-3}{6} to lowest terms by extracting and canceling out 3.

$$\sqrt{5}-\frac{1}{2}\sqrt{7}-\frac{1}{9}\sqrt{180}$$

Factor 180=6^{2}\times 5. Rewrite the square root of the product \sqrt{6^{2}\times 5} as the product of square roots \sqrt{6^{2}}\sqrt{5}. Take the square root of 6^{2}.

$$\sqrt{5}-\frac{1}{2}\sqrt{7}-\frac{1}{9}\times 6\sqrt{5}$$

Express -\frac{1}{9}\times 6 as a single fraction.

$$\sqrt{5}-\frac{1}{2}\sqrt{7}+\frac{-6}{9}\sqrt{5}$$

Reduce the fraction \frac{-6}{9} to lowest terms by extracting and canceling out 3.

$$\sqrt{5}-\frac{1}{2}\sqrt{7}-\frac{2}{3}\sqrt{5}$$

Combine \sqrt{5} and -\frac{2}{3}\sqrt{5} to get \frac{1}{3}\sqrt{5}.

$$\frac{1}{3}\sqrt{5}-\frac{1}{2}\sqrt{7}$$

Factor

$$\frac{2 \sqrt{5} - 3 \sqrt{7}}{6} = -0.5775196630323655$$