Solve (sqrt{7}+3/2-1/sqrt{7}):(4sqrt{7}+7)=

$$( \sqrt { 7 } + \frac { 3 } { 2 } - \frac { 1 } { \sqrt { 7 } } ) : ( 4 \sqrt { 7 } + 7 ) =$$

Evaluate

$$\frac{3}{14}\approx 0.214285714$$

Solution Steps

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

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

The square of \sqrt{7} is 7.

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

Combine \sqrt{7} and -\frac{\sqrt{7}}{7} to get \frac{6}{7}\sqrt{7}.

$$\frac{\frac{6}{7}\sqrt{7}+\frac{3}{2}}{4\sqrt{7}+7}$$

Rationalize the denominator of \frac{\frac{6}{7}\sqrt{7}+\frac{3}{2}}{4\sqrt{7}+7} by multiplying numerator and denominator by 4\sqrt{7}-7.

$$\frac{\left(\frac{6}{7}\sqrt{7}+\frac{3}{2}\right)\left(4\sqrt{7}-7\right)}{\left(4\sqrt{7}+7\right)\left(4\sqrt{7}-7\right)}$$

Consider \left(4\sqrt{7}+7\right)\left(4\sqrt{7}-7\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\frac{6}{7}\sqrt{7}+\frac{3}{2}\right)\left(4\sqrt{7}-7\right)}{\left(4\sqrt{7}\right)^{2}-7^{2}}$$

Expand \left(4\sqrt{7}\right)^{2}.

$$\frac{\left(\frac{6}{7}\sqrt{7}+\frac{3}{2}\right)\left(4\sqrt{7}-7\right)}{4^{2}\left(\sqrt{7}\right)^{2}-7^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(\frac{6}{7}\sqrt{7}+\frac{3}{2}\right)\left(4\sqrt{7}-7\right)}{16\left(\sqrt{7}\right)^{2}-7^{2}}$$

The square of \sqrt{7} is 7.

$$\frac{\left(\frac{6}{7}\sqrt{7}+\frac{3}{2}\right)\left(4\sqrt{7}-7\right)}{16\times 7-7^{2}}$$

Multiply 16 and 7 to get 112.

$$\frac{\left(\frac{6}{7}\sqrt{7}+\frac{3}{2}\right)\left(4\sqrt{7}-7\right)}{112-7^{2}}$$

Calculate 7 to the power of 2 and get 49.

$$\frac{\left(\frac{6}{7}\sqrt{7}+\frac{3}{2}\right)\left(4\sqrt{7}-7\right)}{112-49}$$

Subtract 49 from 112 to get 63.

$$\frac{\left(\frac{6}{7}\sqrt{7}+\frac{3}{2}\right)\left(4\sqrt{7}-7\right)}{63}$$

Apply the distributive property by multiplying each term of \frac{6}{7}\sqrt{7}+\frac{3}{2} by each term of 4\sqrt{7}-7.

$$\frac{\frac{6}{7}\sqrt{7}\times 4\sqrt{7}+\frac{6}{7}\sqrt{7}\left(-7\right)+\frac{3}{2}\times 4\sqrt{7}+\frac{3}{2}\left(-7\right)}{63}$$

Multiply \sqrt{7} and \sqrt{7} to get 7.

$$\frac{\frac{6}{7}\times 7\times 4+\frac{6}{7}\sqrt{7}\left(-7\right)+\frac{3}{2}\times 4\sqrt{7}+\frac{3}{2}\left(-7\right)}{63}$$

Cancel out 7 and 7.

$$\frac{6\times 4+\frac{6}{7}\sqrt{7}\left(-7\right)+\frac{3}{2}\times 4\sqrt{7}+\frac{3}{2}\left(-7\right)}{63}$$

Multiply 6 and 4 to get 24.

$$\frac{24+\frac{6}{7}\sqrt{7}\left(-7\right)+\frac{3}{2}\times 4\sqrt{7}+\frac{3}{2}\left(-7\right)}{63}$$

Express \frac{6}{7}\left(-7\right) as a single fraction.

$$\frac{24+\frac{6\left(-7\right)}{7}\sqrt{7}+\frac{3}{2}\times 4\sqrt{7}+\frac{3}{2}\left(-7\right)}{63}$$

Multiply 6 and -7 to get -42.

$$\frac{24+\frac{-42}{7}\sqrt{7}+\frac{3}{2}\times 4\sqrt{7}+\frac{3}{2}\left(-7\right)}{63}$$

Divide -42 by 7 to get -6.

$$\frac{24-6\sqrt{7}+\frac{3}{2}\times 4\sqrt{7}+\frac{3}{2}\left(-7\right)}{63}$$

Express \frac{3}{2}\times 4 as a single fraction.

$$\frac{24-6\sqrt{7}+\frac{3\times 4}{2}\sqrt{7}+\frac{3}{2}\left(-7\right)}{63}$$

Multiply 3 and 4 to get 12.

$$\frac{24-6\sqrt{7}+\frac{12}{2}\sqrt{7}+\frac{3}{2}\left(-7\right)}{63}$$

Divide 12 by 2 to get 6.

$$\frac{24-6\sqrt{7}+6\sqrt{7}+\frac{3}{2}\left(-7\right)}{63}$$

Combine -6\sqrt{7} and 6\sqrt{7} to get 0.

$$\frac{24+\frac{3}{2}\left(-7\right)}{63}$$

Express \frac{3}{2}\left(-7\right) as a single fraction.

$$\frac{24+\frac{3\left(-7\right)}{2}}{63}$$

Multiply 3 and -7 to get -21.

$$\frac{24+\frac{-21}{2}}{63}$$

Fraction \frac{-21}{2} can be rewritten as -\frac{21}{2} by extracting the negative sign.

$$\frac{24-\frac{21}{2}}{63}$$

Convert 24 to fraction \frac{48}{2}.

$$\frac{\frac{48}{2}-\frac{21}{2}}{63}$$

Since \frac{48}{2} and \frac{21}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{48-21}{2}}{63}$$

Subtract 21 from 48 to get 27.

$$\frac{\frac{27}{2}}{63}$$

Express \frac{\frac{27}{2}}{63} as a single fraction.

$$\frac{27}{2\times 63}$$

Multiply 2 and 63 to get 126.

$$\frac{27}{126}$$

Reduce the fraction \frac{27}{126} to lowest terms by extracting and canceling out 9.

$$\frac{3}{14}$$

Factor

$$\frac{3}{2 \cdot 7} = 0.21428571428571427$$