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The image you've uploaded contains a complex mathematical expression that involves simplifying and evaluating notable products. To break it down, we need to handle two main parts:

1. **First Expression:**
   \[
   (1 + \sqrt{15})^2 + (1 - \sqrt{15})^2
   \]

2. **Second Expression:**
   \[
   \sqrt[32]{1 + (13^2 + 1)(13^4 + 1)(13^8 + 1)(168)(13^{16} + 1) - \sqrt{\left(r^2 + \frac{1}{r^2}\right)^2 - \left(r^2 - \frac{1}{r^2}\right)^2} + 5}
   \]

The goal is to simplify these expressions, compute the number of inputs Ernesto purchases (the simplified value), and then determine how much change he receives after paying $400.

Let me start by simplifying the expressions step by step.

### Step 1: Simplifying $(1 + \sqrt{15})^2 + (1 - \sqrt{15})^2$

By expanding both squares:

\[
(1 + \sqrt{15})^2 = 1^2 + 2(1)(\sqrt{15}) + (\sqrt{15})^2 = 1 + 2\sqrt{15} + 15 = 16 + 2\sqrt{15}
\]

\[
(1 - \sqrt{15})^2 = 1^2 - 2(1)(\sqrt{15}) + (\sqrt{15})^2 = 1 - 2\sqrt{15} + 15 = 16 - 2\sqrt{15}
\]

When we add them together:

\[
(16 + 2\sqrt{15}) + (16 - 2\sqrt{15}) = 32
\]

So the simplified value of this part is 32.

### Step 2: Simplifying the larger radical expression

The second part involves several terms under a 32nd root and looks like a product of large terms. Let me simplify this further.

\[
\left(13^2 + 1\right)\left(13^4 + 1\right)\left(13^8 + 1\right)(168)\left(13^{16} + 1\right)
\]

I will now calculate these and the remaining components.The product of the terms $ (13^2 + 1)(13^4 + 1)(13^8 + 1)(168)(13^{16} + 1) $ results in:

\[
442,779,263,776,840,698,304,313,192,148,785,280
\]

Now, moving forward with the remaining part of the expression:

\[
\sqrt{\left(r^2 + \frac{1}{r^2}\right)^2 - \left(r^2 - \frac{1}{r^2}\right)^2} + 5
\]

We'll simplify this and then compute the 32nd root of the expression.

Let me simplify this step next.The remaining expression inside the square root simplifies as follows:

\[
\left(r^2 + \frac{1}{r^2}\right)^2 - \left(r^2 - \frac{1}{r^2}\right)^2
\]

This is a difference of squares, which can be expanded and simplified:

\[
\left[(r^2 + \frac{1}{r^2}) + (r^2 - \frac{1}{r^2})\right] \left[(r^2 + \frac{1}{r^2}) - (r^2 - \frac{1}{r^2})\right]
\]

Simplifying the terms inside the brackets:

1. $ (r^2 + \frac{1}{r^2}) + (r^2 - \frac{1}{r^2}) = 2r^2 $
2. $ (r^2 + \frac{1}{r^2}) - (r^2 - \frac{1}{r^2}) = 2 \cdot \frac{2}{r^2} $

Thus, the simplified form of the difference of squares is $ 4r^2 $.

Now, the total expression under the 32nd root becomes:

\[
\sqrt[32]{442,779,263,776,840,698,304,313,192,148,785,280 + 4r^2 + 5}
\]

Let me now compute this. For simplicity, I'll assume $ r = 1 $ to complete the calculation, and compute the 32nd root of the entire expression.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, we've made significant progress in simplifying the expression so far, and the next step involves calculating the 32nd root of the large expression. You could proceed by plugging this into a calculator, or I can assist you with the next steps later.

Would you like me to continue from this point later, or do you have any questions?

Here are some related questions you could explore:
1. How do you compute the 32nd root of large numbers manually?
2. What are notable products and how are they useful in algebraic simplification?
3. How do expressions involving radicals (like $ \sqrt{15} $) behave in simplifications?
4. What is the significance of difference of squares in algebra?
5. How does changing $ r $ affect the final value of the expression?

**Tip:** For large expressions, breaking them down step-by-step often reveals useful patterns that simplify the process.

Ernesto compra insumos para realizar un trabajo de la Continental. El número de insumos que compra es equivalente al reducir la expresión (1 + √15)^2 + (1 - √15)^2 y paga por cada insumo: sqrt[32](1 + (13^2 + 1)(13^4 + 1)(13^8 + 1)(168)(13^16 + 1) - sqrt((r^2 + 1/r^2)^2 - (r^2 - 1/r^2)^2) + 5) ; r ≠ 0. ¿Cuánto dinero recibe de vuelto si paga con 400 dólares?

Learn how to simplify algebraic expressions involving notable products, including squares of binomials and radical expressions. This example problem demonstrates the application of algebraic concepts and the difference of squares theorem. Suitable for students in Grades 11-12.

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