Evaluate
\sqrt{170}+5\sqrt{7}-\sqrt{238}-7\sqrt{5}\approx -4.812563097
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\frac{\left(\sqrt{5}-\sqrt{7}\right)\left(\sqrt{35}+\sqrt{34}\right)}{\left(\sqrt{35}-\sqrt{34}\right)\left(\sqrt{35}+\sqrt{34}\right)}
Rationalize the denominator of \frac{\sqrt{5}-\sqrt{7}}{\sqrt{35}-\sqrt{34}} by multiplying numerator and denominator by \sqrt{35}+\sqrt{34}.
\frac{\left(\sqrt{5}-\sqrt{7}\right)\left(\sqrt{35}+\sqrt{34}\right)}{\left(\sqrt{35}\right)^{2}-\left(\sqrt{34}\right)^{2}}
Consider \left(\sqrt{35}-\sqrt{34}\right)\left(\sqrt{35}+\sqrt{34}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
\frac{\left(\sqrt{5}-\sqrt{7}\right)\left(\sqrt{35}+\sqrt{34}\right)}{35-34}
Square \sqrt{35}. Square \sqrt{34}.
\frac{\left(\sqrt{5}-\sqrt{7}\right)\left(\sqrt{35}+\sqrt{34}\right)}{1}
Subtract 34 from 35 to get 1.
\left(\sqrt{5}-\sqrt{7}\right)\left(\sqrt{35}+\sqrt{34}\right)
Anything divided by one gives itself.
\sqrt{5}\sqrt{35}+\sqrt{5}\sqrt{34}-\sqrt{7}\sqrt{35}-\sqrt{7}\sqrt{34}
Apply the distributive property by multiplying each term of \sqrt{5}-\sqrt{7} by each term of \sqrt{35}+\sqrt{34}.
\sqrt{5}\sqrt{5}\sqrt{7}+\sqrt{5}\sqrt{34}-\sqrt{7}\sqrt{35}-\sqrt{7}\sqrt{34}
Factor 35=5\times 7. Rewrite the square root of the product \sqrt{5\times 7} as the product of square roots \sqrt{5}\sqrt{7}.
5\sqrt{7}+\sqrt{5}\sqrt{34}-\sqrt{7}\sqrt{35}-\sqrt{7}\sqrt{34}
Multiply \sqrt{5} and \sqrt{5} to get 5.
5\sqrt{7}+\sqrt{170}-\sqrt{7}\sqrt{35}-\sqrt{7}\sqrt{34}
To multiply \sqrt{5} and \sqrt{34}, multiply the numbers under the square root.
5\sqrt{7}+\sqrt{170}-\sqrt{7}\sqrt{7}\sqrt{5}-\sqrt{7}\sqrt{34}
Factor 35=7\times 5. Rewrite the square root of the product \sqrt{7\times 5} as the product of square roots \sqrt{7}\sqrt{5}.
5\sqrt{7}+\sqrt{170}-7\sqrt{5}-\sqrt{7}\sqrt{34}
Multiply \sqrt{7} and \sqrt{7} to get 7.
5\sqrt{7}+\sqrt{170}-7\sqrt{5}-\sqrt{238}
To multiply \sqrt{7} and \sqrt{34}, multiply the numbers under the square root.
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