Solve frac{2sqrt{7}-1}{2sqrt{7}+5}

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

Evaluate

$$11-4\sqrt{7}\approx 0.416994756$$

Solution Steps

Rationalize the denominator of \frac{2\sqrt{7}-1}{2\sqrt{7}+5} by multiplying numerator and denominator by 2\sqrt{7}-5.

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

Consider \left(2\sqrt{7}+5\right)\left(2\sqrt{7}-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}.

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

Expand \left(2\sqrt{7}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{7} is 7.

$$\frac{\left(2\sqrt{7}-1\right)\left(2\sqrt{7}-5\right)}{4\times 7-5^{2}}$$

Multiply 4 and 7 to get 28.

$$\frac{\left(2\sqrt{7}-1\right)\left(2\sqrt{7}-5\right)}{28-5^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{\left(2\sqrt{7}-1\right)\left(2\sqrt{7}-5\right)}{28-25}$$

Subtract 25 from 28 to get 3.

$$\frac{\left(2\sqrt{7}-1\right)\left(2\sqrt{7}-5\right)}{3}$$

Apply the distributive property by multiplying each term of 2\sqrt{7}-1 by each term of 2\sqrt{7}-5.

$$\frac{4\left(\sqrt{7}\right)^{2}-10\sqrt{7}-2\sqrt{7}+5}{3}$$

The square of \sqrt{7} is 7.

$$\frac{4\times 7-10\sqrt{7}-2\sqrt{7}+5}{3}$$

Multiply 4 and 7 to get 28.

$$\frac{28-10\sqrt{7}-2\sqrt{7}+5}{3}$$

Combine -10\sqrt{7} and -2\sqrt{7} to get -12\sqrt{7}.

$$\frac{28-12\sqrt{7}+5}{3}$$

Add 28 and 5 to get 33.

$$\frac{33-12\sqrt{7}}{3}$$

Divide each term of 33-12\sqrt{7} by 3 to get 11-4\sqrt{7}.

$$11-4\sqrt{7}$$