Solve x-x*45/82^23-sqrt{2*2div5}

$$x-x \times \frac{ 45 }{ { 82 }^{ 2 } 3- \sqrt{ 2 \times 2 \div 5 } }$$

Factor

$$\frac{\left(1015004608-45\sqrt{5}\right)x}{1017273958}$$

Solution Steps

Calculate 82 to the power of 2 and get 6724.

$$factor(x-x\times \frac{45}{6724\times 3-\sqrt{\frac{2\times 2}{5}}})$$

Multiply 6724 and 3 to get 20172.

$$factor(x-x\times \frac{45}{20172-\sqrt{\frac{2\times 2}{5}}})$$

Multiply 2 and 2 to get 4.

$$factor(x-x\times \frac{45}{20172-\sqrt{\frac{4}{5}}})$$

Rewrite the square root of the division \sqrt{\frac{4}{5}} as the division of square roots \frac{\sqrt{4}}{\sqrt{5}}.

$$factor(x-x\times \frac{45}{20172-\frac{\sqrt{4}}{\sqrt{5}}})$$

Calculate the square root of 4 and get 2.

$$factor(x-x\times \frac{45}{20172-\frac{2}{\sqrt{5}}})$$

Rationalize the denominator of \frac{2}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$factor(x-x\times \frac{45}{20172-\frac{2\sqrt{5}}{\left(\sqrt{5}\right)^{2}}})$$

The square of \sqrt{5} is 5.

$$factor(x-x\times \frac{45}{20172-\frac{2\sqrt{5}}{5}})$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 20172 times \frac{5}{5}.

$$factor(x-x\times \frac{45}{\frac{20172\times 5}{5}-\frac{2\sqrt{5}}{5}})$$

Since \frac{20172\times 5}{5} and \frac{2\sqrt{5}}{5} have the same denominator, subtract them by subtracting their numerators.

$$factor(x-x\times \frac{45}{\frac{20172\times 5-2\sqrt{5}}{5}})$$

Do the multiplications in 20172\times 5-2\sqrt{5}.

$$factor(x-x\times \frac{45}{\frac{100860-2\sqrt{5}}{5}})$$

Divide 45 by \frac{100860-2\sqrt{5}}{5} by multiplying 45 by the reciprocal of \frac{100860-2\sqrt{5}}{5}.

$$factor(x-x\times \frac{45\times 5}{100860-2\sqrt{5}})$$

Rationalize the denominator of \frac{45\times 5}{100860-2\sqrt{5}} by multiplying numerator and denominator by 100860+2\sqrt{5}.

$$factor(x-x\times \frac{45\times 5\left(100860+2\sqrt{5}\right)}{\left(100860-2\sqrt{5}\right)\left(100860+2\sqrt{5}\right)})$$

Consider \left(100860-2\sqrt{5}\right)\left(100860+2\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$factor(x-x\times \frac{45\times 5\left(100860+2\sqrt{5}\right)}{100860^{2}-\left(-2\sqrt{5}\right)^{2}})$$

Multiply 45 and 5 to get 225.

$$factor(x-x\times \frac{225\left(100860+2\sqrt{5}\right)}{100860^{2}-\left(-2\sqrt{5}\right)^{2}})$$

Calculate 100860 to the power of 2 and get 10172739600.

$$factor(x-x\times \frac{225\left(100860+2\sqrt{5}\right)}{10172739600-\left(-2\sqrt{5}\right)^{2}})$$

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

$$factor(x-x\times \frac{225\left(100860+2\sqrt{5}\right)}{10172739600-\left(-2\right)^{2}\left(\sqrt{5}\right)^{2}})$$

Calculate -2 to the power of 2 and get 4.

$$factor(x-x\times \frac{225\left(100860+2\sqrt{5}\right)}{10172739600-4\left(\sqrt{5}\right)^{2}})$$

The square of \sqrt{5} is 5.

$$factor(x-x\times \frac{225\left(100860+2\sqrt{5}\right)}{10172739600-4\times 5})$$

Multiply 4 and 5 to get 20.

$$factor(x-x\times \frac{225\left(100860+2\sqrt{5}\right)}{10172739600-20})$$

Subtract 20 from 10172739600 to get 10172739580.

$$factor(x-x\times \frac{225\left(100860+2\sqrt{5}\right)}{10172739580})$$

Divide 225\left(100860+2\sqrt{5}\right) by 10172739580 to get \frac{45}{2034547916}\left(100860+2\sqrt{5}\right).

$$factor(x-x\times \frac{45}{2034547916}\left(100860+2\sqrt{5}\right))$$

Use the distributive property to multiply \frac{45}{2034547916} by 100860+2\sqrt{5}.

$$factor(x-x\left(\frac{45}{2034547916}\times 100860+\frac{45}{2034547916}\times 2\sqrt{5}\right))$$

Express \frac{45}{2034547916}\times 100860 as a single fraction.

$$factor(x-x\left(\frac{45\times 100860}{2034547916}+\frac{45}{2034547916}\times 2\sqrt{5}\right))$$

Multiply 45 and 100860 to get 4538700.

$$factor(x-x\left(\frac{4538700}{2034547916}+\frac{45}{2034547916}\times 2\sqrt{5}\right))$$

Reduce the fraction \frac{4538700}{2034547916} to lowest terms by extracting and canceling out 4.

$$factor(x-x\left(\frac{1134675}{508636979}+\frac{45}{2034547916}\times 2\sqrt{5}\right))$$

Express \frac{45}{2034547916}\times 2 as a single fraction.

$$factor(x-x\left(\frac{1134675}{508636979}+\frac{45\times 2}{2034547916}\sqrt{5}\right))$$

Multiply 45 and 2 to get 90.

$$factor(x-x\left(\frac{1134675}{508636979}+\frac{90}{2034547916}\sqrt{5}\right))$$

Reduce the fraction \frac{90}{2034547916} to lowest terms by extracting and canceling out 2.

$$factor(x-x\left(\frac{1134675}{508636979}+\frac{45}{1017273958}\sqrt{5}\right))$$

Use the distributive property to multiply x by \frac{1134675}{508636979}+\frac{45}{1017273958}\sqrt{5}.

$$factor(x-\left(x\times \frac{1134675}{508636979}+x\times \frac{45}{1017273958}\sqrt{5}\right))$$

To find the opposite of x\times \frac{1134675}{508636979}+x\times \frac{45}{1017273958}\sqrt{5}, find the opposite of each term.

$$factor(x-x\times \frac{1134675}{508636979}-x\times \frac{45}{1017273958}\sqrt{5})$$

Multiply -1 and \frac{1134675}{508636979} to get -\frac{1134675}{508636979}.

$$factor(x-\frac{1134675}{508636979}x-x\times \frac{45}{1017273958}\sqrt{5})$$

Combine x and -\frac{1134675}{508636979}x to get \frac{507502304}{508636979}x.

$$factor(\frac{507502304}{508636979}x-x\times \frac{45}{1017273958}\sqrt{5})$$

Multiply -1 and \frac{45}{1017273958} to get -\frac{45}{1017273958}.

$$factor(\frac{507502304}{508636979}x-\frac{45}{1017273958}x\sqrt{5})$$

Factor out \frac{1}{1017273958}.

$$\frac{1015004608x-45x\sqrt{5}}{1017273958}$$

Consider 1015004608x-45x\sqrt{5}. Factor out x.

$$x\left(1015004608-45\sqrt{5}\right)$$

Rewrite the complete factored expression.

$$\frac{x\left(1015004608-45\sqrt{5}\right)}{1017273958}$$

Evaluate

$$\frac{\left(1015004608-45\sqrt{5}\right)x}{1017273958}$$