The expression shown in the image is:

$\frac{7}{7^{-1}} + 7^{-4} \cdot (-7)^4$

Let's simplify it step by step.

Step 1: Simplify $\frac{7}{7^{-1}}$

Using the property of exponents $a^{-n} = \frac{1}{a^n}$, we know that:

$7^{-1} = \frac{1}{7}$

So,

$\frac{7}{7^{-1}} = 7 \times 7 = 49$

Step 2: Simplify $7^{-4} \cdot (-7)^4$

• $7^{-4}$ means $\frac{1}{7^4} = \frac{1}{2401}$
• $(-7)^4$ means $(-7)^4 = 2401$, since raising a negative number to an even power results in a positive value.

Thus, we have:

$7^{-4} \cdot (-7)^4 = \frac{1}{2401} \cdot 2401 = 1$

Step 3: Add the results

Now, adding the two simplified parts:

$49 + 1 = 50$

Final Answer:

The value of the expression is $50$.

Would you like a deeper explanation or details on any specific step?

Here are 5 related questions for further exploration:

1. How would the expression change if the base 7 was replaced with another number?
2. What happens when you apply the same process to fractional exponents?
3. Can negative exponents always be simplified using the same rules?
4. How do the properties of exponents behave with multiplication of negative numbers?
5. What happens if odd powers are applied to negative bases?

Tip: Always keep track of exponent properties, especially when dealing with negative numbers and fractional exponents. They can dramatically change the outcome!