https://r2.erweima.ai/mathbot/image/1721144909588sotez4j0.png

Luego como el cubo tiene 6 caras, se multiplica por 6, resultando el área 6a al cuadrado 

V3= (ab-b al cuadrado)*a=a al cuadrado b - a b al cuadrado 
Sumando los 3 volumenes tendriamos 
a al cubo - a al cuadrado b+ a b al cuadrado + a al cuadrado b- a b al cuadrado = a al cubo 

Let's analyze the problem and the image step by step to understand the solution provided.

### Analysis of the Image
The image depicts a 3D shape composed of a larger cube from which a smaller cube has been removed. Let's denote:
- \( a \) as the side length of the smaller cube.
- \( b \) as the side length of the larger cube.

### Area Calculation
The explanation states that since the cube has 6 faces, its total surface area is \( 6a^2 \). This suggests that we are considering the surface area of the smaller cube. 

### Volume Calculation
To find the total volume of the remaining shape, the volume of the smaller cube is subtracted from the volume of the larger cube.

1. **Volume of the larger cube**: \( V_1 = b^3 \)
2. **Volume of the smaller cube**: \( V_2 = a^3 \)

### Calculating the Volumes of the Removed Parts
The parts removed are in the shape of rectangular prisms (cuboids). To find these volumes:

- The height of each cuboid is \( b - a \) (since the side of the smaller cube is removed from the larger one).
- The dimensions of the cuboids are given by the arrangement in the image.

### Total Volume Calculation
From the text, it states:
\[ V_3 = (ab - b^2) \cdot a = a^2b - ab^2 \]

Summing up all the volumes:
\[ a^3 - a^2b + ab^2 + a^2b - ab^2 = a^3 \]

### Summarizing
The total volume of the original shape (before removing the smaller cube) is indeed \( a^3 \).

Would you like a more detailed explanation or have any specific questions?

Here are five questions you might ask:
1. How do we derive the formula for the surface area of the smaller cube?
2. Why do we subtract the volume of the smaller cube from the larger one?
3. What is the significance of each term in the volume summation?
4. How can we generalize this problem for other geometric shapes?
5. Can we apply this method to different configurations of cubes?

**Tip:** When dealing with 3D geometry problems, always start by identifying and labeling all relevant dimensions and volumes before proceeding with calculations.

Explore the relationship between the volumes and surface areas of cubes. This page details the calculations involved in determining the surface area of a smaller cube and the total volume after subtracting the smaller cube from a larger one. Perfect for high school students.

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