Solve (-6/49b-12/35b^2+6b^3):(6/7b)-(2/3b+1)*(frac{4}{5}b-frac{4}{5})+frac{4}{5}b*(1+frac{2}{3})

$$( - \frac { 6 } { 49 } b - \frac { 12 } { 35 } b ^ { 2 } + 6 b ^ { 3 } ) : ( \frac { 6 } { 7 } b ) - ( \frac { 2 } { 3 } b + 1 ) \cdot ( \frac { 4 } { 5 } b - \frac { 4 } { 5 } ) + \frac { 4 } { 5 } b \cdot ( 1 + \frac { 2 } { 3 } )$$

Evaluate

$$\frac{97b^{2}}{15}+\frac{2b}{3}+\frac{23}{35}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{-\frac{6}{49}b-\frac{12}{35}b^{2}+6b^{3}}{\frac{6}{7}b}.

$$\frac{\frac{6}{245}\times 245b\left(b-\left(-\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)\left(b-\left(\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)}{\frac{6}{7}b}-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Cancel out b in both numerator and denominator.

$$\frac{\frac{6}{245}\times 245\left(b-\left(-\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)\left(b-\left(\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)}{\frac{6}{7}}-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Multiply \frac{6}{245} and 245 to get 6.

$$\frac{6\left(b-\left(-\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)\left(b-\left(\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)}{\frac{6}{7}}-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Divide 6\left(b-\left(-\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)\left(b-\left(\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right) by \frac{6}{7} to get 7\left(b-\left(-\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)\left(b-\left(\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right).

$$7\left(b-\left(-\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)\left(b-\left(\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Use the distributive property to multiply 7 by b-\left(-\frac{1}{35}\sqrt{26}+\frac{1}{35}\right).

$$\left(7b+\frac{1}{5}\sqrt{26}-\frac{1}{5}\right)\left(b-\left(\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

To find the opposite of \frac{1}{35}\sqrt{26}+\frac{1}{35}, find the opposite of each term.

$$\left(7b+\frac{1}{5}\sqrt{26}-\frac{1}{5}\right)\left(b-\frac{1}{35}\sqrt{26}-\frac{1}{35}\right)-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Use the distributive property to multiply 7b+\frac{1}{5}\sqrt{26}-\frac{1}{5} by b-\frac{1}{35}\sqrt{26}-\frac{1}{35} and combine like terms.

$$7b^{2}-\frac{2}{5}b-\frac{1}{175}\left(\sqrt{26}\right)^{2}+\frac{1}{175}-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

The square of \sqrt{26} is 26.

$$7b^{2}-\frac{2}{5}b-\frac{1}{175}\times 26+\frac{1}{175}-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Multiply -\frac{1}{175} and 26 to get -\frac{26}{175}.

$$7b^{2}-\frac{2}{5}b-\frac{26}{175}+\frac{1}{175}-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Add -\frac{26}{175} and \frac{1}{175} to get -\frac{1}{7}.

$$7b^{2}-\frac{2}{5}b-\frac{1}{7}-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Use the distributive property to multiply \frac{2}{3}b+1 by \frac{4}{5}b-\frac{4}{5} and combine like terms.

$$7b^{2}-\frac{2}{5}b-\frac{1}{7}-\left(\frac{8}{15}b^{2}+\frac{4}{15}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

To find the opposite of \frac{8}{15}b^{2}+\frac{4}{15}b-\frac{4}{5}, find the opposite of each term.

$$7b^{2}-\frac{2}{5}b-\frac{1}{7}-\frac{8}{15}b^{2}-\frac{4}{15}b+\frac{4}{5}+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Combine 7b^{2} and -\frac{8}{15}b^{2} to get \frac{97}{15}b^{2}.

$$\frac{97}{15}b^{2}-\frac{2}{5}b-\frac{1}{7}-\frac{4}{15}b+\frac{4}{5}+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Combine -\frac{2}{5}b and -\frac{4}{15}b to get -\frac{2}{3}b.

$$\frac{97}{15}b^{2}-\frac{2}{3}b-\frac{1}{7}+\frac{4}{5}+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Add -\frac{1}{7} and \frac{4}{5} to get \frac{23}{35}.

$$\frac{97}{15}b^{2}-\frac{2}{3}b+\frac{23}{35}+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Add 1 and \frac{2}{3} to get \frac{5}{3}.

$$\frac{97}{15}b^{2}-\frac{2}{3}b+\frac{23}{35}+\frac{4}{5}b\times \frac{5}{3}$$

Multiply \frac{4}{5} and \frac{5}{3} to get \frac{4}{3}.

$$\frac{97}{15}b^{2}-\frac{2}{3}b+\frac{23}{35}+\frac{4}{3}b$$

Combine -\frac{2}{3}b and \frac{4}{3}b to get \frac{2}{3}b.

$$\frac{97}{15}b^{2}+\frac{2}{3}b+\frac{23}{35}$$

Expand

$$\frac{97b^{2}}{15}+\frac{2b}{3}+\frac{23}{35}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{-\frac{6}{49}b-\frac{12}{35}b^{2}+6b^{3}}{\frac{6}{7}b}.

$$\frac{\frac{6}{245}\times 245b\left(b-\left(-\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)\left(b-\left(\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)}{\frac{6}{7}b}-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Cancel out b in both numerator and denominator.

$$\frac{\frac{6}{245}\times 245\left(b-\left(-\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)\left(b-\left(\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)}{\frac{6}{7}}-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Multiply \frac{6}{245} and 245 to get 6.

$$\frac{6\left(b-\left(-\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)\left(b-\left(\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)}{\frac{6}{7}}-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Divide 6\left(b-\left(-\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)\left(b-\left(\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right) by \frac{6}{7} to get 7\left(b-\left(-\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)\left(b-\left(\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right).

$$7\left(b-\left(-\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)\left(b-\left(\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Use the distributive property to multiply 7 by b-\left(-\frac{1}{35}\sqrt{26}+\frac{1}{35}\right).

$$\left(7b+\frac{1}{5}\sqrt{26}-\frac{1}{5}\right)\left(b-\left(\frac{1}{35}\sqrt{26}+\frac{1}{35}\right)\right)-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

To find the opposite of \frac{1}{35}\sqrt{26}+\frac{1}{35}, find the opposite of each term.

$$\left(7b+\frac{1}{5}\sqrt{26}-\frac{1}{5}\right)\left(b-\frac{1}{35}\sqrt{26}-\frac{1}{35}\right)-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Use the distributive property to multiply 7b+\frac{1}{5}\sqrt{26}-\frac{1}{5} by b-\frac{1}{35}\sqrt{26}-\frac{1}{35} and combine like terms.

$$7b^{2}-\frac{2}{5}b-\frac{1}{175}\left(\sqrt{26}\right)^{2}+\frac{1}{175}-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

The square of \sqrt{26} is 26.

$$7b^{2}-\frac{2}{5}b-\frac{1}{175}\times 26+\frac{1}{175}-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Multiply -\frac{1}{175} and 26 to get -\frac{26}{175}.

$$7b^{2}-\frac{2}{5}b-\frac{26}{175}+\frac{1}{175}-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Add -\frac{26}{175} and \frac{1}{175} to get -\frac{1}{7}.

$$7b^{2}-\frac{2}{5}b-\frac{1}{7}-\left(\frac{2}{3}b+1\right)\left(\frac{4}{5}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Use the distributive property to multiply \frac{2}{3}b+1 by \frac{4}{5}b-\frac{4}{5} and combine like terms.

$$7b^{2}-\frac{2}{5}b-\frac{1}{7}-\left(\frac{8}{15}b^{2}+\frac{4}{15}b-\frac{4}{5}\right)+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

To find the opposite of \frac{8}{15}b^{2}+\frac{4}{15}b-\frac{4}{5}, find the opposite of each term.

$$7b^{2}-\frac{2}{5}b-\frac{1}{7}-\frac{8}{15}b^{2}-\frac{4}{15}b+\frac{4}{5}+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Combine 7b^{2} and -\frac{8}{15}b^{2} to get \frac{97}{15}b^{2}.

$$\frac{97}{15}b^{2}-\frac{2}{5}b-\frac{1}{7}-\frac{4}{15}b+\frac{4}{5}+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Combine -\frac{2}{5}b and -\frac{4}{15}b to get -\frac{2}{3}b.

$$\frac{97}{15}b^{2}-\frac{2}{3}b-\frac{1}{7}+\frac{4}{5}+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Add -\frac{1}{7} and \frac{4}{5} to get \frac{23}{35}.

$$\frac{97}{15}b^{2}-\frac{2}{3}b+\frac{23}{35}+\frac{4}{5}b\left(1+\frac{2}{3}\right)$$

Add 1 and \frac{2}{3} to get \frac{5}{3}.

$$\frac{97}{15}b^{2}-\frac{2}{3}b+\frac{23}{35}+\frac{4}{5}b\times \frac{5}{3}$$

Multiply \frac{4}{5} and \frac{5}{3} to get \frac{4}{3}.

$$\frac{97}{15}b^{2}-\frac{2}{3}b+\frac{23}{35}+\frac{4}{3}b$$

Combine -\frac{2}{3}b and \frac{4}{3}b to get \frac{2}{3}b.

$$\frac{97}{15}b^{2}+\frac{2}{3}b+\frac{23}{35}$$