Solve 2(a^2+1/a^2)-(a-1/a)-7

$$2 ( a ^ { 2 } + \frac { 1 } { a ^ { 2 } } ) - ( a - \frac { 1 } { a } ) - 7$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply a^{2} times \frac{a^{2}}{a^{2}}.

$$2\left(\frac{a^{2}a^{2}}{a^{2}}+\frac{1}{a^{2}}\right)-\left(a-\frac{1}{a}\right)-7$$

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

$$2\times \frac{a^{2}a^{2}+1}{a^{2}}-\left(a-\frac{1}{a}\right)-7$$

Do the multiplications in a^{2}a^{2}+1.

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

Express 2\times \frac{a^{4}+1}{a^{2}} as a single fraction.

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

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

$$\frac{2\left(a^{4}+1\right)}{a^{2}}-\left(\frac{aa}{a}-\frac{1}{a}\right)-7$$

Since \frac{aa}{a} and \frac{1}{a} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(a^{4}+1\right)}{a^{2}}-\frac{aa-1}{a}-7$$

Do the multiplications in aa-1.

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

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

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

Since \frac{2\left(a^{4}+1\right)}{a^{2}} and \frac{\left(a^{2}-1\right)a}{a^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2\left(a^{4}+1\right)-\left(a^{2}-1\right)a.

$$\frac{2a^{4}+2-a^{3}+a}{a^{2}}-7$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 7 times \frac{a^{2}}{a^{2}}.

$$\frac{2a^{4}+2-a^{3}+a}{a^{2}}-\frac{7a^{2}}{a^{2}}$$

Since \frac{2a^{4}+2-a^{3}+a}{a^{2}} and \frac{7a^{2}}{a^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2a^{4}+2-a^{3}+a-7a^{2}}{a^{2}}$$

Expand

$$2a^{2}-a-7+\frac{1}{a}+\frac{2}{a^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply a^{2} times \frac{a^{2}}{a^{2}}.

$$2\left(\frac{a^{2}a^{2}}{a^{2}}+\frac{1}{a^{2}}\right)-\left(a-\frac{1}{a}\right)-7$$

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

$$2\times \frac{a^{2}a^{2}+1}{a^{2}}-\left(a-\frac{1}{a}\right)-7$$

Do the multiplications in a^{2}a^{2}+1.

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

Express 2\times \frac{a^{4}+1}{a^{2}} as a single fraction.

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

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

$$\frac{2\left(a^{4}+1\right)}{a^{2}}-\left(\frac{aa}{a}-\frac{1}{a}\right)-7$$

Since \frac{aa}{a} and \frac{1}{a} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(a^{4}+1\right)}{a^{2}}-\frac{aa-1}{a}-7$$

Do the multiplications in aa-1.

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

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

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

Since \frac{2\left(a^{4}+1\right)}{a^{2}} and \frac{\left(a^{2}-1\right)a}{a^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2\left(a^{4}+1\right)-\left(a^{2}-1\right)a.

$$\frac{2a^{4}+2-a^{3}+a}{a^{2}}-7$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 7 times \frac{a^{2}}{a^{2}}.

$$\frac{2a^{4}+2-a^{3}+a}{a^{2}}-\frac{7a^{2}}{a^{2}}$$

Since \frac{2a^{4}+2-a^{3}+a}{a^{2}} and \frac{7a^{2}}{a^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2a^{4}+2-a^{3}+a-7a^{2}}{a^{2}}$$