Solve E=frac{(5-sqrt{24})(sqrt{75}+sqrt{50})}{sqrt{75}-sqrt{50}}

$$E = \frac { ( 5 - \sqrt { 24 } ) ( \sqrt { 75 } + \sqrt { 50 } ) } { \sqrt { 75 } - \sqrt { 50 } }$$

Solve for E

$$E=1$$

Steps for Solving Linear Equation

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(\sqrt{75}+\sqrt{50}\right)}{\sqrt{75}-\sqrt{50}}$$

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(5\sqrt{3}+\sqrt{50}\right)}{\sqrt{75}-\sqrt{50}}$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(5\sqrt{3}+5\sqrt{2}\right)}{\sqrt{75}-\sqrt{50}}$$

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(5\sqrt{3}+5\sqrt{2}\right)}{5\sqrt{3}-\sqrt{50}}$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(5\sqrt{3}+5\sqrt{2}\right)}{5\sqrt{3}-5\sqrt{2}}$$

Rationalize the denominator of \frac{\left(5-2\sqrt{6}\right)\left(5\sqrt{3}+5\sqrt{2}\right)}{5\sqrt{3}-5\sqrt{2}} by multiplying numerator and denominator by 5\sqrt{3}+5\sqrt{2}.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(5\sqrt{3}+5\sqrt{2}\right)\left(5\sqrt{3}+5\sqrt{2}\right)}{\left(5\sqrt{3}-5\sqrt{2}\right)\left(5\sqrt{3}+5\sqrt{2}\right)}$$

Consider \left(5\sqrt{3}-5\sqrt{2}\right)\left(5\sqrt{3}+5\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(5\sqrt{3}+5\sqrt{2}\right)\left(5\sqrt{3}+5\sqrt{2}\right)}{\left(5\sqrt{3}\right)^{2}-\left(-5\sqrt{2}\right)^{2}}$$

Multiply 5\sqrt{3}+5\sqrt{2} and 5\sqrt{3}+5\sqrt{2} to get \left(5\sqrt{3}+5\sqrt{2}\right)^{2}.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(5\sqrt{3}+5\sqrt{2}\right)^{2}}{\left(5\sqrt{3}\right)^{2}-\left(-5\sqrt{2}\right)^{2}}$$

Expand \left(5\sqrt{3}\right)^{2}.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(5\sqrt{3}+5\sqrt{2}\right)^{2}}{5^{2}\left(\sqrt{3}\right)^{2}-\left(-5\sqrt{2}\right)^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(5\sqrt{3}+5\sqrt{2}\right)^{2}}{25\left(\sqrt{3}\right)^{2}-\left(-5\sqrt{2}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(5\sqrt{3}+5\sqrt{2}\right)^{2}}{25\times 3-\left(-5\sqrt{2}\right)^{2}}$$

Multiply 25 and 3 to get 75.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(5\sqrt{3}+5\sqrt{2}\right)^{2}}{75-\left(-5\sqrt{2}\right)^{2}}$$

Expand \left(-5\sqrt{2}\right)^{2}.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(5\sqrt{3}+5\sqrt{2}\right)^{2}}{75-\left(-5\right)^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate -5 to the power of 2 and get 25.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(5\sqrt{3}+5\sqrt{2}\right)^{2}}{75-25\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(5\sqrt{3}+5\sqrt{2}\right)^{2}}{75-25\times 2}$$

Multiply 25 and 2 to get 50.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(5\sqrt{3}+5\sqrt{2}\right)^{2}}{75-50}$$

Subtract 50 from 75 to get 25.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(5\sqrt{3}+5\sqrt{2}\right)^{2}}{25}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5\sqrt{3}+5\sqrt{2}\right)^{2}.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(25\left(\sqrt{3}\right)^{2}+50\sqrt{3}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)}{25}$$

The square of \sqrt{3} is 3.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(25\times 3+50\sqrt{3}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)}{25}$$

Multiply 25 and 3 to get 75.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(75+50\sqrt{3}\sqrt{2}+25\left(\sqrt{2}\right)^{2}\right)}{25}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(75+50\sqrt{6}+25\left(\sqrt{2}\right)^{2}\right)}{25}$$

The square of \sqrt{2} is 2.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(75+50\sqrt{6}+25\times 2\right)}{25}$$

Multiply 25 and 2 to get 50.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(75+50\sqrt{6}+50\right)}{25}$$

Add 75 and 50 to get 125.

$$E=\frac{\left(5-2\sqrt{6}\right)\left(125+50\sqrt{6}\right)}{25}$$

Apply the distributive property by multiplying each term of 5-2\sqrt{6} by each term of 125+50\sqrt{6}.

$$E=\frac{625+250\sqrt{6}-250\sqrt{6}-100\left(\sqrt{6}\right)^{2}}{25}$$

Combine 250\sqrt{6} and -250\sqrt{6} to get 0.

$$E=\frac{625-100\left(\sqrt{6}\right)^{2}}{25}$$

The square of \sqrt{6} is 6.

$$E=\frac{625-100\times 6}{25}$$

Multiply -100 and 6 to get -600.

$$E=\frac{625-600}{25}$$

Subtract 600 from 625 to get 25.

$$E=\frac{25}{25}$$

Divide 25 by 25 to get 1.

$$E=1$$

Assign E

$$E≔1$$