Solve 48divsqrt{28}-68divsqrt{45}-2sqrt{20}+2sqrt{175}

$$48 \div \sqrt{ 28 } -68 \div \sqrt{ 45 } -2 \sqrt{ 20 } +2 \sqrt{ 175 }$$

Evaluate

$$\frac{94\sqrt{7}}{7}-\frac{128\sqrt{5}}{15}\approx 16.447547055$$

Solution Steps

Factor 28=2^{2}\times 7. Rewrite the square root of the product \sqrt{2^{2}\times 7} as the product of square roots \sqrt{2^{2}}\sqrt{7}. Take the square root of 2^{2}.

$$\frac{48}{2\sqrt{7}}-\frac{68}{\sqrt{45}}-2\sqrt{20}+2\sqrt{175}$$

Rationalize the denominator of \frac{48}{2\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{48\sqrt{7}}{2\left(\sqrt{7}\right)^{2}}-\frac{68}{\sqrt{45}}-2\sqrt{20}+2\sqrt{175}$$

The square of \sqrt{7} is 7.

$$\frac{48\sqrt{7}}{2\times 7}-\frac{68}{\sqrt{45}}-2\sqrt{20}+2\sqrt{175}$$

Cancel out 2 in both numerator and denominator.

$$\frac{24\sqrt{7}}{7}-\frac{68}{\sqrt{45}}-2\sqrt{20}+2\sqrt{175}$$

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}.

$$\frac{24\sqrt{7}}{7}-\frac{68}{3\sqrt{5}}-2\sqrt{20}+2\sqrt{175}$$

Rationalize the denominator of \frac{68}{3\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{24\sqrt{7}}{7}-\frac{68\sqrt{5}}{3\left(\sqrt{5}\right)^{2}}-2\sqrt{20}+2\sqrt{175}$$

The square of \sqrt{5} is 5.

$$\frac{24\sqrt{7}}{7}-\frac{68\sqrt{5}}{3\times 5}-2\sqrt{20}+2\sqrt{175}$$

Multiply 3 and 5 to get 15.

$$\frac{24\sqrt{7}}{7}-\frac{68\sqrt{5}}{15}-2\sqrt{20}+2\sqrt{175}$$

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\frac{24\sqrt{7}}{7}-\frac{68\sqrt{5}}{15}-2\times 2\sqrt{5}+2\sqrt{175}$$

Multiply -2 and 2 to get -4.

$$\frac{24\sqrt{7}}{7}-\frac{68\sqrt{5}}{15}-4\sqrt{5}+2\sqrt{175}$$

Combine -\frac{68\sqrt{5}}{15} and -4\sqrt{5} to get -\frac{128}{15}\sqrt{5}.

$$\frac{24\sqrt{7}}{7}-\frac{128}{15}\sqrt{5}+2\sqrt{175}$$

Factor 175=5^{2}\times 7. Rewrite the square root of the product \sqrt{5^{2}\times 7} as the product of square roots \sqrt{5^{2}}\sqrt{7}. Take the square root of 5^{2}.

$$\frac{24\sqrt{7}}{7}-\frac{128}{15}\sqrt{5}+2\times 5\sqrt{7}$$

Multiply 2 and 5 to get 10.

$$\frac{24\sqrt{7}}{7}-\frac{128}{15}\sqrt{5}+10\sqrt{7}$$

Combine \frac{24\sqrt{7}}{7} and 10\sqrt{7} to get \frac{94}{7}\sqrt{7}.

$$\frac{94}{7}\sqrt{7}-\frac{128}{15}\sqrt{5}$$

Factor

$$\frac{2 {(705 \sqrt{7} - 448 \sqrt{5})}}{105} = 16.447547054869155$$