Solve (7/8a-5/6b)(3a+1/7b)

$$( \frac { 7 } { 8 } a - \frac { 5 } { 6 } b ) ( 3 a + \frac { 1 } { 7 } b )$$

Evaluate

$$\frac{\left(21a-20b\right)\left(21a+b\right)}{168}$$

Solution Steps

Apply the distributive property by multiplying each term of \frac{7}{8}a-\frac{5}{6}b by each term of 3a+\frac{1}{7}b.

$$\frac{7}{8}a\times 3a+\frac{7}{8}a\times \frac{1}{7}b-\frac{5}{6}b\times 3a-\frac{5}{6}b\times \frac{1}{7}b$$

Multiply a and a to get a^{2}.

$$\frac{7}{8}a^{2}\times 3+\frac{7}{8}a\times \frac{1}{7}b-\frac{5}{6}b\times 3a-\frac{5}{6}b\times \frac{1}{7}b$$

Multiply b and b to get b^{2}.

$$\frac{7}{8}a^{2}\times 3+\frac{7}{8}a\times \frac{1}{7}b-\frac{5}{6}b\times 3a-\frac{5}{6}b^{2}\times \frac{1}{7}$$

Express \frac{7}{8}\times 3 as a single fraction.

$$\frac{7\times 3}{8}a^{2}+\frac{7}{8}a\times \frac{1}{7}b-\frac{5}{6}b\times 3a-\frac{5}{6}b^{2}\times \frac{1}{7}$$

Multiply 7 and 3 to get 21.

$$\frac{21}{8}a^{2}+\frac{7}{8}a\times \frac{1}{7}b-\frac{5}{6}b\times 3a-\frac{5}{6}b^{2}\times \frac{1}{7}$$

Multiply \frac{7}{8} times \frac{1}{7} by multiplying numerator times numerator and denominator times denominator.

$$\frac{21}{8}a^{2}+\frac{7\times 1}{8\times 7}ab-\frac{5}{6}b\times 3a-\frac{5}{6}b^{2}\times \frac{1}{7}$$

Cancel out 7 in both numerator and denominator.

$$\frac{21}{8}a^{2}+\frac{1}{8}ab-\frac{5}{6}b\times 3a-\frac{5}{6}b^{2}\times \frac{1}{7}$$

Express -\frac{5}{6}\times 3 as a single fraction.

$$\frac{21}{8}a^{2}+\frac{1}{8}ab+\frac{-5\times 3}{6}ba-\frac{5}{6}b^{2}\times \frac{1}{7}$$

Multiply -5 and 3 to get -15.

$$\frac{21}{8}a^{2}+\frac{1}{8}ab+\frac{-15}{6}ba-\frac{5}{6}b^{2}\times \frac{1}{7}$$

Reduce the fraction \frac{-15}{6} to lowest terms by extracting and canceling out 3.

$$\frac{21}{8}a^{2}+\frac{1}{8}ab-\frac{5}{2}ba-\frac{5}{6}b^{2}\times \frac{1}{7}$$

Combine \frac{1}{8}ab and -\frac{5}{2}ba to get -\frac{19}{8}ab.

$$\frac{21}{8}a^{2}-\frac{19}{8}ab-\frac{5}{6}b^{2}\times \frac{1}{7}$$

Multiply -\frac{5}{6} times \frac{1}{7} by multiplying numerator times numerator and denominator times denominator.

$$\frac{21}{8}a^{2}-\frac{19}{8}ab+\frac{-5}{6\times 7}b^{2}$$

Do the multiplications in the fraction \frac{-5}{6\times 7}.

$$\frac{21}{8}a^{2}-\frac{19}{8}ab+\frac{-5}{42}b^{2}$$

Fraction \frac{-5}{42} can be rewritten as -\frac{5}{42} by extracting the negative sign.

$$\frac{21}{8}a^{2}-\frac{19}{8}ab-\frac{5}{42}b^{2}$$

Expand

$$-\frac{19ab}{8}+\frac{21a^{2}}{8}-\frac{5b^{2}}{42}$$

Solution Steps

Apply the distributive property by multiplying each term of \frac{7}{8}a-\frac{5}{6}b by each term of 3a+\frac{1}{7}b.

$$\frac{7}{8}a\times 3a+\frac{7}{8}a\times \frac{1}{7}b-\frac{5}{6}b\times 3a-\frac{5}{6}b\times \frac{1}{7}b$$

Multiply a and a to get a^{2}.

$$\frac{7}{8}a^{2}\times 3+\frac{7}{8}a\times \frac{1}{7}b-\frac{5}{6}b\times 3a-\frac{5}{6}b\times \frac{1}{7}b$$

Multiply b and b to get b^{2}.

$$\frac{7}{8}a^{2}\times 3+\frac{7}{8}a\times \frac{1}{7}b-\frac{5}{6}b\times 3a-\frac{5}{6}b^{2}\times \frac{1}{7}$$

Express \frac{7}{8}\times 3 as a single fraction.

$$\frac{7\times 3}{8}a^{2}+\frac{7}{8}a\times \frac{1}{7}b-\frac{5}{6}b\times 3a-\frac{5}{6}b^{2}\times \frac{1}{7}$$

Multiply 7 and 3 to get 21.

$$\frac{21}{8}a^{2}+\frac{7}{8}a\times \frac{1}{7}b-\frac{5}{6}b\times 3a-\frac{5}{6}b^{2}\times \frac{1}{7}$$

Multiply \frac{7}{8} times \frac{1}{7} by multiplying numerator times numerator and denominator times denominator.

$$\frac{21}{8}a^{2}+\frac{7\times 1}{8\times 7}ab-\frac{5}{6}b\times 3a-\frac{5}{6}b^{2}\times \frac{1}{7}$$

Cancel out 7 in both numerator and denominator.

$$\frac{21}{8}a^{2}+\frac{1}{8}ab-\frac{5}{6}b\times 3a-\frac{5}{6}b^{2}\times \frac{1}{7}$$

Express -\frac{5}{6}\times 3 as a single fraction.

$$\frac{21}{8}a^{2}+\frac{1}{8}ab+\frac{-5\times 3}{6}ba-\frac{5}{6}b^{2}\times \frac{1}{7}$$

Multiply -5 and 3 to get -15.

$$\frac{21}{8}a^{2}+\frac{1}{8}ab+\frac{-15}{6}ba-\frac{5}{6}b^{2}\times \frac{1}{7}$$

Reduce the fraction \frac{-15}{6} to lowest terms by extracting and canceling out 3.

$$\frac{21}{8}a^{2}+\frac{1}{8}ab-\frac{5}{2}ba-\frac{5}{6}b^{2}\times \frac{1}{7}$$

Combine \frac{1}{8}ab and -\frac{5}{2}ba to get -\frac{19}{8}ab.

$$\frac{21}{8}a^{2}-\frac{19}{8}ab-\frac{5}{6}b^{2}\times \frac{1}{7}$$

Multiply -\frac{5}{6} times \frac{1}{7} by multiplying numerator times numerator and denominator times denominator.

$$\frac{21}{8}a^{2}-\frac{19}{8}ab+\frac{-5}{6\times 7}b^{2}$$

Do the multiplications in the fraction \frac{-5}{6\times 7}.

$$\frac{21}{8}a^{2}-\frac{19}{8}ab+\frac{-5}{42}b^{2}$$

Fraction \frac{-5}{42} can be rewritten as -\frac{5}{42} by extracting the negative sign.

$$\frac{21}{8}a^{2}-\frac{19}{8}ab-\frac{5}{42}b^{2}$$