Solve 1/4sqrt{80}-1/16sqrt{63}-1/9sqrt{180}

$$\frac { 1 } { 4 } \sqrt { 80 } - \frac { 1 } { 16 } \sqrt { 63 } - \frac { 1 } { 9 } \sqrt { 180 }$$

Evaluate

$$\frac{\sqrt{5}}{3}-\frac{3\sqrt{7}}{16}\approx 0.249277622$$

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}.

$$\frac{1}{4}\times 4\sqrt{5}-\frac{1}{16}\sqrt{63}-\frac{1}{9}\sqrt{180}$$

Cancel out 4 and 4.

$$\sqrt{5}-\frac{1}{16}\sqrt{63}-\frac{1}{9}\sqrt{180}$$

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}.

$$\sqrt{5}-\frac{1}{16}\times 3\sqrt{7}-\frac{1}{9}\sqrt{180}$$

Express -\frac{1}{16}\times 3 as a single fraction.

$$\sqrt{5}+\frac{-3}{16}\sqrt{7}-\frac{1}{9}\sqrt{180}$$

Fraction \frac{-3}{16} can be rewritten as -\frac{3}{16} by extracting the negative sign.

$$\sqrt{5}-\frac{3}{16}\sqrt{7}-\frac{1}{9}\sqrt{180}$$

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}.

$$\sqrt{5}-\frac{3}{16}\sqrt{7}-\frac{1}{9}\times 6\sqrt{5}$$

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

$$\sqrt{5}-\frac{3}{16}\sqrt{7}+\frac{-6}{9}\sqrt{5}$$

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

$$\sqrt{5}-\frac{3}{16}\sqrt{7}-\frac{2}{3}\sqrt{5}$$

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

$$\frac{1}{3}\sqrt{5}-\frac{3}{16}\sqrt{7}$$

Factor

$$\frac{16 \sqrt{5} - 9 \sqrt{7}}{48} = 0.2492776216753192$$