Solve 5sqrt{700}-4sqrt{343}-3sqrt{112}-21sqrt{7^-1}

$$5 \sqrt{ 700 } -4 \sqrt{ 343 } -3 \sqrt{ 112 } -21 \sqrt{ { 7 }^{ -1 } }$$

Evaluate

$$7\sqrt{7}\approx 18.520259177$$

Solution Steps

Factor 700=10^{2}\times 7. Rewrite the square root of the product \sqrt{10^{2}\times 7} as the product of square roots \sqrt{10^{2}}\sqrt{7}. Take the square root of 10^{2}.

$$5\times 10\sqrt{7}-4\sqrt{343}-3\sqrt{112}-21\sqrt{7^{-1}}$$

Multiply 5 and 10 to get 50.

$$50\sqrt{7}-4\sqrt{343}-3\sqrt{112}-21\sqrt{7^{-1}}$$

Factor 343=7^{2}\times 7. Rewrite the square root of the product \sqrt{7^{2}\times 7} as the product of square roots \sqrt{7^{2}}\sqrt{7}. Take the square root of 7^{2}.

$$50\sqrt{7}-4\times 7\sqrt{7}-3\sqrt{112}-21\sqrt{7^{-1}}$$

Multiply -4 and 7 to get -28.

$$50\sqrt{7}-28\sqrt{7}-3\sqrt{112}-21\sqrt{7^{-1}}$$

Combine 50\sqrt{7} and -28\sqrt{7} to get 22\sqrt{7}.

$$22\sqrt{7}-3\sqrt{112}-21\sqrt{7^{-1}}$$

Factor 112=4^{2}\times 7. Rewrite the square root of the product \sqrt{4^{2}\times 7} as the product of square roots \sqrt{4^{2}}\sqrt{7}. Take the square root of 4^{2}.

$$22\sqrt{7}-3\times 4\sqrt{7}-21\sqrt{7^{-1}}$$

Multiply -3 and 4 to get -12.

$$22\sqrt{7}-12\sqrt{7}-21\sqrt{7^{-1}}$$

Combine 22\sqrt{7} and -12\sqrt{7} to get 10\sqrt{7}.

$$10\sqrt{7}-21\sqrt{7^{-1}}$$

Calculate 7 to the power of -1 and get \frac{1}{7}.

$$10\sqrt{7}-21\sqrt{\frac{1}{7}}$$

Rewrite the square root of the division \sqrt{\frac{1}{7}} as the division of square roots \frac{\sqrt{1}}{\sqrt{7}}.

$$10\sqrt{7}-21\times \frac{\sqrt{1}}{\sqrt{7}}$$

Calculate the square root of 1 and get 1.

$$10\sqrt{7}-21\times \frac{1}{\sqrt{7}}$$

Rationalize the denominator of \frac{1}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$10\sqrt{7}-21\times \frac{\sqrt{7}}{\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$10\sqrt{7}-21\times \frac{\sqrt{7}}{7}$$

Cancel out 7, the greatest common factor in 21 and 7.

$$10\sqrt{7}-3\sqrt{7}$$

Combine 10\sqrt{7} and -3\sqrt{7} to get 7\sqrt{7}.

$$7\sqrt{7}$$