Solve -{(1+1/a)*(1+1/a+1/a^2)-(3(a-1)/a^2)}

$$- \{ ( 1 + \frac { 1 } { a } ) \times ( 1 + \frac { 1 } { a } + \frac { 1 } { a ^ { 2 } } ) - ( \frac { 3 ( a - 1 ) } { a ^ { 2 } } ) \}$$

Evaluate

$$-1+\frac{1}{a}-\frac{5}{a^{2}}-\frac{1}{a^{3}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a}{a}.

$$-\left(\left(\frac{a}{a}+\frac{1}{a}\right)\left(1+\frac{1}{a}+\frac{1}{a^{2}}\right)-\frac{3\left(a-1\right)}{a^{2}}\right)$$

Since \frac{a}{a} and \frac{1}{a} have the same denominator, add them by adding their numerators.

$$-\left(\frac{a+1}{a}\left(1+\frac{1}{a}+\frac{1}{a^{2}}\right)-\frac{3\left(a-1\right)}{a^{2}}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a}{a}.

$$-\left(\frac{a+1}{a}\left(\frac{a}{a}+\frac{1}{a}+\frac{1}{a^{2}}\right)-\frac{3\left(a-1\right)}{a^{2}}\right)$$

Since \frac{a}{a} and \frac{1}{a} have the same denominator, add them by adding their numerators.

$$-\left(\frac{a+1}{a}\left(\frac{a+1}{a}+\frac{1}{a^{2}}\right)-\frac{3\left(a-1\right)}{a^{2}}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a and a^{2} is a^{2}. Multiply \frac{a+1}{a} times \frac{a}{a}.

$$-\left(\frac{a+1}{a}\left(\frac{\left(a+1\right)a}{a^{2}}+\frac{1}{a^{2}}\right)-\frac{3\left(a-1\right)}{a^{2}}\right)$$

Since \frac{\left(a+1\right)a}{a^{2}} and \frac{1}{a^{2}} have the same denominator, add them by adding their numerators.

$$-\left(\frac{a+1}{a}\times \frac{\left(a+1\right)a+1}{a^{2}}-\frac{3\left(a-1\right)}{a^{2}}\right)$$

Do the multiplications in \left(a+1\right)a+1.

$$-\left(\frac{a+1}{a}\times \frac{a^{2}+a+1}{a^{2}}-\frac{3\left(a-1\right)}{a^{2}}\right)$$

Multiply \frac{a+1}{a} times \frac{a^{2}+a+1}{a^{2}} by multiplying numerator times numerator and denominator times denominator.

$$-\left(\frac{\left(a+1\right)\left(a^{2}+a+1\right)}{aa^{2}}-\frac{3\left(a-1\right)}{a^{2}}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of aa^{2} and a^{2} is aa^{2}. Multiply \frac{3\left(a-1\right)}{a^{2}} times \frac{a}{a}.

$$-\left(\frac{\left(a+1\right)\left(a^{2}+a+1\right)}{aa^{2}}-\frac{3\left(a-1\right)a}{aa^{2}}\right)$$

Since \frac{\left(a+1\right)\left(a^{2}+a+1\right)}{aa^{2}} and \frac{3\left(a-1\right)a}{aa^{2}} have the same denominator, subtract them by subtracting their numerators.

$$-\frac{\left(a+1\right)\left(a^{2}+a+1\right)-3\left(a-1\right)a}{aa^{2}}$$

Do the multiplications in \left(a+1\right)\left(a^{2}+a+1\right)-3\left(a-1\right)a.

$$-\frac{a^{3}+a^{2}+a+a^{2}+a+1-3a^{2}+3a}{aa^{2}}$$

Combine like terms in a^{3}+a^{2}+a+a^{2}+a+1-3a^{2}+3a.

$$-\frac{a^{3}-a^{2}+5a+1}{aa^{2}}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$-\frac{a^{3}-a^{2}+5a+1}{a^{3}}$$

Expand

$$-1+\frac{1}{a}-\frac{5}{a^{2}}-\frac{1}{a^{3}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a}{a}.

$$-\left(\left(\frac{a}{a}+\frac{1}{a}\right)\left(1+\frac{1}{a}+\frac{1}{a^{2}}\right)-\frac{3\left(a-1\right)}{a^{2}}\right)$$

Since \frac{a}{a} and \frac{1}{a} have the same denominator, add them by adding their numerators.

$$-\left(\frac{a+1}{a}\left(1+\frac{1}{a}+\frac{1}{a^{2}}\right)-\frac{3\left(a-1\right)}{a^{2}}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{a}{a}.

$$-\left(\frac{a+1}{a}\left(\frac{a}{a}+\frac{1}{a}+\frac{1}{a^{2}}\right)-\frac{3\left(a-1\right)}{a^{2}}\right)$$

Since \frac{a}{a} and \frac{1}{a} have the same denominator, add them by adding their numerators.

$$-\left(\frac{a+1}{a}\left(\frac{a+1}{a}+\frac{1}{a^{2}}\right)-\frac{3\left(a-1\right)}{a^{2}}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a and a^{2} is a^{2}. Multiply \frac{a+1}{a} times \frac{a}{a}.

$$-\left(\frac{a+1}{a}\left(\frac{\left(a+1\right)a}{a^{2}}+\frac{1}{a^{2}}\right)-\frac{3\left(a-1\right)}{a^{2}}\right)$$

Since \frac{\left(a+1\right)a}{a^{2}} and \frac{1}{a^{2}} have the same denominator, add them by adding their numerators.

$$-\left(\frac{a+1}{a}\times \frac{\left(a+1\right)a+1}{a^{2}}-\frac{3\left(a-1\right)}{a^{2}}\right)$$

Do the multiplications in \left(a+1\right)a+1.

$$-\left(\frac{a+1}{a}\times \frac{a^{2}+a+1}{a^{2}}-\frac{3\left(a-1\right)}{a^{2}}\right)$$

Multiply \frac{a+1}{a} times \frac{a^{2}+a+1}{a^{2}} by multiplying numerator times numerator and denominator times denominator.

$$-\left(\frac{\left(a+1\right)\left(a^{2}+a+1\right)}{aa^{2}}-\frac{3\left(a-1\right)}{a^{2}}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of aa^{2} and a^{2} is aa^{2}. Multiply \frac{3\left(a-1\right)}{a^{2}} times \frac{a}{a}.

$$-\left(\frac{\left(a+1\right)\left(a^{2}+a+1\right)}{aa^{2}}-\frac{3\left(a-1\right)a}{aa^{2}}\right)$$

Since \frac{\left(a+1\right)\left(a^{2}+a+1\right)}{aa^{2}} and \frac{3\left(a-1\right)a}{aa^{2}} have the same denominator, subtract them by subtracting their numerators.

$$-\frac{\left(a+1\right)\left(a^{2}+a+1\right)-3\left(a-1\right)a}{aa^{2}}$$

Do the multiplications in \left(a+1\right)\left(a^{2}+a+1\right)-3\left(a-1\right)a.

$$-\frac{a^{3}+a^{2}+a+a^{2}+a+1-3a^{2}+3a}{aa^{2}}$$

Combine like terms in a^{3}+a^{2}+a+a^{2}+a+1-3a^{2}+3a.

$$-\frac{a^{3}-a^{2}+5a+1}{aa^{2}}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$-\frac{a^{3}-a^{2}+5a+1}{a^{3}}$$