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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

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

Evaluate

$a^{4}-\frac{1}{a^{4}}$

Short Solution Steps

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

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

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

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

Do the multiplications in $aa-1$.

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

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

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

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

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

Do the multiplications in $aa+1$.

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

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

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

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

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

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

$$\frac{a^{2}-1}{a}\times \frac{a^{2}+1}{a}\times \frac{a^{4}+1}{a^{2}}$$

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

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

Multiply $\frac{\left(a^{2}-1\right)\left(a^{2}+1\right)}{aa}$ times $\frac{a^{4}+1}{a^{2}}$ by multiplying numerator times numerator and denominator times denominator.

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

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

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

To multiply powers of the same base, add their exponents. Add $2$ and $2$ to get $4$.

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

Use the distributive property to multiply $a^{2}-1$ by $a^{2}+1$ and combine like terms.

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

Consider $\left(a^{4}-1\right)\left(a^{4}+1\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$. Square $1$.

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

To raise a power to another power, multiply the exponents. Multiply $4$ and $2$ to get $8$.

$$\frac{a^{8}-1}{a^{4}}$$

Expand

$a^{4}-\frac{1}{a^{4}}$

Short Solution Steps

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

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

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

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

Do the multiplications in $aa-1$.

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

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

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

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

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

Do the multiplications in $aa+1$.

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

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

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

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

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

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

$$\frac{a^{2}-1}{a}\times \frac{a^{2}+1}{a}\times \frac{a^{4}+1}{a^{2}}$$

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

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

Multiply $\frac{\left(a^{2}-1\right)\left(a^{2}+1\right)}{aa}$ times $\frac{a^{4}+1}{a^{2}}$ by multiplying numerator times numerator and denominator times denominator.

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

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

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

To multiply powers of the same base, add their exponents. Add $2$ and $2$ to get $4$.

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

Use the distributive property to multiply $a^{2}-1$ by $a^{2}+1$ and combine like terms.

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

Consider $\left(a^{4}-1\right)\left(a^{4}+1\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$. Square $1$.

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

To raise a power to another power, multiply the exponents. Multiply $4$ and $2$ to get $8$.

$$\frac{a^{8}-1}{a^{4}}$$