Solve frac{sqrt{23}+5}{2}+2/2(sqrt{23+5)}

$$\frac { \sqrt { 23 } + 5 } { 2 } + \frac { 2 } { 2 ( \sqrt { 23 } + 5 ) }$$

Evaluate

$$5$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 2\left(\sqrt{23}+5\right) is 2\left(\sqrt{23}+5\right). Multiply \frac{\sqrt{23}+5}{2} times \frac{\sqrt{23}+5}{\sqrt{23}+5}.

$$\frac{\left(\sqrt{23}+5\right)\left(\sqrt{23}+5\right)}{2\left(\sqrt{23}+5\right)}+\frac{2}{2\left(\sqrt{23}+5\right)}$$

Since \frac{\left(\sqrt{23}+5\right)\left(\sqrt{23}+5\right)}{2\left(\sqrt{23}+5\right)} and \frac{2}{2\left(\sqrt{23}+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(\sqrt{23}+5\right)\left(\sqrt{23}+5\right)+2}{2\left(\sqrt{23}+5\right)}$$

Do the multiplications in \left(\sqrt{23}+5\right)\left(\sqrt{23}+5\right)+2.

$$\frac{23+5\sqrt{23}+5\sqrt{23}+25+2}{2\left(\sqrt{23}+5\right)}$$

Do the calculations in 23+5\sqrt{23}+5\sqrt{23}+25+2.

$$\frac{50+10\sqrt{23}}{2\left(\sqrt{23}+5\right)}$$

Expand 2\left(\sqrt{23}+5\right).

$$\frac{50+10\sqrt{23}}{2\sqrt{23}+10}$$

Rationalize the denominator of \frac{50+10\sqrt{23}}{2\sqrt{23}+10} by multiplying numerator and denominator by 2\sqrt{23}-10.

$$\frac{\left(50+10\sqrt{23}\right)\left(2\sqrt{23}-10\right)}{\left(2\sqrt{23}+10\right)\left(2\sqrt{23}-10\right)}$$

Consider \left(2\sqrt{23}+10\right)\left(2\sqrt{23}-10\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(50+10\sqrt{23}\right)\left(2\sqrt{23}-10\right)}{\left(2\sqrt{23}\right)^{2}-10^{2}}$$

Expand \left(2\sqrt{23}\right)^{2}.

$$\frac{\left(50+10\sqrt{23}\right)\left(2\sqrt{23}-10\right)}{2^{2}\left(\sqrt{23}\right)^{2}-10^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(50+10\sqrt{23}\right)\left(2\sqrt{23}-10\right)}{4\left(\sqrt{23}\right)^{2}-10^{2}}$$

The square of \sqrt{23} is 23.

$$\frac{\left(50+10\sqrt{23}\right)\left(2\sqrt{23}-10\right)}{4\times 23-10^{2}}$$

Multiply 4 and 23 to get 92.

$$\frac{\left(50+10\sqrt{23}\right)\left(2\sqrt{23}-10\right)}{92-10^{2}}$$

Calculate 10 to the power of 2 and get 100.

$$\frac{\left(50+10\sqrt{23}\right)\left(2\sqrt{23}-10\right)}{92-100}$$

Subtract 100 from 92 to get -8.

$$\frac{\left(50+10\sqrt{23}\right)\left(2\sqrt{23}-10\right)}{-8}$$

Apply the distributive property by multiplying each term of 50+10\sqrt{23} by each term of 2\sqrt{23}-10.

$$\frac{100\sqrt{23}-500+20\left(\sqrt{23}\right)^{2}-100\sqrt{23}}{-8}$$

The square of \sqrt{23} is 23.

$$\frac{100\sqrt{23}-500+20\times 23-100\sqrt{23}}{-8}$$

Multiply 20 and 23 to get 460.

$$\frac{100\sqrt{23}-500+460-100\sqrt{23}}{-8}$$

Add -500 and 460 to get -40.

$$\frac{100\sqrt{23}-40-100\sqrt{23}}{-8}$$

Combine 100\sqrt{23} and -100\sqrt{23} to get 0.

$$\frac{-40}{-8}$$

Divide -40 by -8 to get 5.

$$5$$

Factor

$$5$$