f $$ a-\frac{1}{a}=m $$, show that $$ a^{4}+\frac{1}{a^{4}}=m^{4}+4 m^{2}+2 $$

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To show that $$ a^{4} + \frac{1}{a^{4}} = m^{4} + 4m^{2} + 2 $$ given $$ a - \frac{1}{a} = m $$, we can follow these steps:

1. **Start with the given equation:**
   $$
   a - \frac{1}{a} = m
   $$

2. **Square both sides to find $$ a^2 + \frac{1}{a^2} $$:**
   $$
   \left(a - \frac{1}{a}\right)^2 = m^2
   $$
   $$
   a^2 - 2\left(a \cdot \frac{1}{a}\right) + \frac{1}{a^2} = m^2
   $$
   $$
   a^2 + \frac{1}{a^2} - 2 = m^2
   $$
   $$
   a^2 + \frac{1}{a^2} = m^2 + 2
   $$

3. **Square $$ a^2 + \frac{1}{a^2} $$ to find $$ a^4 + \frac{1}{a^4} $$:**
   $$
   \left(a^2 + \frac{1}{a^2}\right)^2 = (m^2 + 2)^2
   $$
   $$
   a^4 + 2\left(a^2 \cdot \frac{1}{a^2}\right) + \frac{1}{a^4} = m^4 + 4m^2 + 4
   $$
   $$
   a^4 + \frac{1}{a^4} + 2 = m^4 + 4m^2 + 4
   $$
   $$
   a^4 + \frac{1}{a^4} = m^4 + 4m^2 + 2
   $$

Thus, we've shown that:
$$
a^{4} + \frac{1}{a^{4}} = m^{4} + 4m^{2} + 2
$$

**Answer:** After expanding, a⁴ + 1/a⁴ is m⁴ plus four m² plus two. Thus,

a⁴ + 1/a⁴ = m⁴ + 4m² + 2
$$ a^{4} + \frac{1}{a^{4}} = m^{4} + 4m^{2} + 2 $$

f , show that

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