Solve ((a + 4)/(a - 4) - (a - 4)/(a + 4))(1/4 + 1/a) | Equatix - Math Solver

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Example

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

a⋅(b-2)=3b

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

a⋅(b-2)=3b

Question

$$(\frac{a+4}{a-4}-\frac{a-4}{a+4})\cdot(\frac{1}{4}+\frac{1}{a})$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $a-4$ and $a+4$ is $\left(a-4\right)\left(a+4\right)$. Multiply $\frac{a+4}{a-4}$ times $\frac{a+4}{a+4}$. Multiply $\frac{a-4}{a+4}$ times $\frac{a-4}{a-4}$.

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

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

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

Do the multiplications in $\left(a+4\right)\left(a+4\right)-\left(a-4\right)\left(a-4\right)$.

$$\frac{a^{2}+4a+4a+16-a^{2}+4a+4a-16}{\left(a-4\right)\left(a+4\right)}\left(\frac{1}{4}+\frac{1}{a}\right)$$

Combine like terms in $a^{2}+4a+4a+16-a^{2}+4a+4a-16$.

$$\frac{16a}{\left(a-4\right)\left(a+4\right)}\left(\frac{1}{4}+\frac{1}{a}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $4$ and $a$ is $4a$. Multiply $\frac{1}{4}$ times $\frac{a}{a}$. Multiply $\frac{1}{a}$ times $\frac{4}{4}$.

$$\frac{16a}{\left(a-4\right)\left(a+4\right)}\left(\frac{a}{4a}+\frac{4}{4a}\right)$$

Since $\frac{a}{4a}$ and $\frac{4}{4a}$ have the same denominator, add them by adding their numerators.

$$\frac{16a}{\left(a-4\right)\left(a+4\right)}\times \frac{a+4}{4a}$$

Multiply $\frac{16a}{\left(a-4\right)\left(a+4\right)}$ times $\frac{a+4}{4a}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{16a\left(a+4\right)}{\left(a-4\right)\left(a+4\right)\times 4a}$$

Cancel out $4a\left(a+4\right)$ in both numerator and denominator.

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $a-4$ and $a+4$ is $\left(a-4\right)\left(a+4\right)$. Multiply $\frac{a+4}{a-4}$ times $\frac{a+4}{a+4}$. Multiply $\frac{a-4}{a+4}$ times $\frac{a-4}{a-4}$.

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

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

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

Do the multiplications in $\left(a+4\right)\left(a+4\right)-\left(a-4\right)\left(a-4\right)$.

$$\frac{a^{2}+4a+4a+16-a^{2}+4a+4a-16}{\left(a-4\right)\left(a+4\right)}\left(\frac{1}{4}+\frac{1}{a}\right)$$

Combine like terms in $a^{2}+4a+4a+16-a^{2}+4a+4a-16$.

$$\frac{16a}{\left(a-4\right)\left(a+4\right)}\left(\frac{1}{4}+\frac{1}{a}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $4$ and $a$ is $4a$. Multiply $\frac{1}{4}$ times $\frac{a}{a}$. Multiply $\frac{1}{a}$ times $\frac{4}{4}$.

$$\frac{16a}{\left(a-4\right)\left(a+4\right)}\left(\frac{a}{4a}+\frac{4}{4a}\right)$$

Since $\frac{a}{4a}$ and $\frac{4}{4a}$ have the same denominator, add them by adding their numerators.

$$\frac{16a}{\left(a-4\right)\left(a+4\right)}\times \frac{a+4}{4a}$$

Multiply $\frac{16a}{\left(a-4\right)\left(a+4\right)}$ times $\frac{a+4}{4a}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{16a\left(a+4\right)}{\left(a-4\right)\left(a+4\right)\times 4a}$$

Cancel out $4a\left(a+4\right)$ in both numerator and denominator.

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