Solve frac{2018sqrt{2018}-1}{2018+sqrt{2018}+1}

$$\frac{ 2018 \sqrt{ 2018 } -1 }{ 2018+ \sqrt{ 2018 } +1 }$$

Evaluate

$$\sqrt{2018}-1\approx 43.92215489$$

Solution Steps

Add 2018 and 1 to get 2019.

$$\frac{2018\sqrt{2018}-1}{2019+\sqrt{2018}}$$

Rationalize the denominator of \frac{2018\sqrt{2018}-1}{2019+\sqrt{2018}} by multiplying numerator and denominator by 2019-\sqrt{2018}.

$$\frac{\left(2018\sqrt{2018}-1\right)\left(2019-\sqrt{2018}\right)}{\left(2019+\sqrt{2018}\right)\left(2019-\sqrt{2018}\right)}$$

Consider \left(2019+\sqrt{2018}\right)\left(2019-\sqrt{2018}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(2018\sqrt{2018}-1\right)\left(2019-\sqrt{2018}\right)}{2019^{2}-\left(\sqrt{2018}\right)^{2}}$$

Square 2019. Square \sqrt{2018}.

$$\frac{\left(2018\sqrt{2018}-1\right)\left(2019-\sqrt{2018}\right)}{4076361-2018}$$

Subtract 2018 from 4076361 to get 4074343.

$$\frac{\left(2018\sqrt{2018}-1\right)\left(2019-\sqrt{2018}\right)}{4074343}$$

Apply the distributive property by multiplying each term of 2018\sqrt{2018}-1 by each term of 2019-\sqrt{2018}.

$$\frac{4074342\sqrt{2018}-2018\left(\sqrt{2018}\right)^{2}-2019+\sqrt{2018}}{4074343}$$

The square of \sqrt{2018} is 2018.

$$\frac{4074342\sqrt{2018}-2018\times 2018-2019+\sqrt{2018}}{4074343}$$

Multiply -2018 and 2018 to get -4072324.

$$\frac{4074342\sqrt{2018}-4072324-2019+\sqrt{2018}}{4074343}$$

Subtract 2019 from -4072324 to get -4074343.

$$\frac{4074342\sqrt{2018}-4074343+\sqrt{2018}}{4074343}$$

Combine 4074342\sqrt{2018} and \sqrt{2018} to get 4074343\sqrt{2018}.

$$\frac{4074343\sqrt{2018}-4074343}{4074343}$$

Divide each term of 4074343\sqrt{2018}-4074343 by 4074343 to get \sqrt{2018}-1.

$$\sqrt{2018}-1$$