Solve 1/4-sqrt{24+1}-1/sqrt{7+24-1}=

$$\frac { 1 } { 4 - \sqrt { 24 } + 1 } - \frac { 1 } { \sqrt { 7 + 24 - 1 } } =$$

Evaluate

$$-\frac{\sqrt{30}}{30}+2\sqrt{6}+5\approx 9.7164053$$

Short Solution Steps

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}.

$$\frac{1}{4-2\sqrt{6}+1}-\frac{1}{\sqrt{7+24-1}}$$

Add 4 and 1 to get 5.

$$\frac{1}{5-2\sqrt{6}}-\frac{1}{\sqrt{7+24-1}}$$

Rationalize the denominator of \frac{1}{5-2\sqrt{6}} by multiplying numerator and denominator by 5+2\sqrt{6}.

$$\frac{5+2\sqrt{6}}{\left(5-2\sqrt{6}\right)\left(5+2\sqrt{6}\right)}-\frac{1}{\sqrt{7+24-1}}$$

Consider \left(5-2\sqrt{6}\right)\left(5+2\sqrt{6}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{5+2\sqrt{6}}{5^{2}-\left(-2\sqrt{6}\right)^{2}}-\frac{1}{\sqrt{7+24-1}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{5+2\sqrt{6}}{25-\left(-2\sqrt{6}\right)^{2}}-\frac{1}{\sqrt{7+24-1}}$$

Expand \left(-2\sqrt{6}\right)^{2}.

$$\frac{5+2\sqrt{6}}{25-\left(-2\right)^{2}\left(\sqrt{6}\right)^{2}}-\frac{1}{\sqrt{7+24-1}}$$

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

$$\frac{5+2\sqrt{6}}{25-4\left(\sqrt{6}\right)^{2}}-\frac{1}{\sqrt{7+24-1}}$$

The square of \sqrt{6} is 6.

$$\frac{5+2\sqrt{6}}{25-4\times 6}-\frac{1}{\sqrt{7+24-1}}$$

Multiply 4 and 6 to get 24.

$$\frac{5+2\sqrt{6}}{25-24}-\frac{1}{\sqrt{7+24-1}}$$

Subtract 24 from 25 to get 1.

$$\frac{5+2\sqrt{6}}{1}-\frac{1}{\sqrt{7+24-1}}$$

Anything divided by one gives itself.

$$5+2\sqrt{6}-\frac{1}{\sqrt{7+24-1}}$$

Add 7 and 24 to get 31.

$$5+2\sqrt{6}-\frac{1}{\sqrt{31-1}}$$

Subtract 1 from 31 to get 30.

$$5+2\sqrt{6}-\frac{1}{\sqrt{30}}$$

Rationalize the denominator of \frac{1}{\sqrt{30}} by multiplying numerator and denominator by \sqrt{30}.

$$5+2\sqrt{6}-\frac{\sqrt{30}}{\left(\sqrt{30}\right)^{2}}$$

The square of \sqrt{30} is 30.

$$5+2\sqrt{6}-\frac{\sqrt{30}}{30}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 5+2\sqrt{6} times \frac{30}{30}.

$$\frac{30\left(5+2\sqrt{6}\right)}{30}-\frac{\sqrt{30}}{30}$$

Since \frac{30\left(5+2\sqrt{6}\right)}{30} and \frac{\sqrt{30}}{30} have the same denominator, subtract them by subtracting their numerators.

$$\frac{30\left(5+2\sqrt{6}\right)-\sqrt{30}}{30}$$

Do the multiplications in 30\left(5+2\sqrt{6}\right)-\sqrt{30}.

$$\frac{150+60\sqrt{6}-\sqrt{30}}{30}$$