Determine the total volume by using the volumes of smaller rectangular prisms.

1. Identify the dimensions for the height, length, and width of each prism.
2. Write an expression for the total volume.

Given expressions:
- [tex]\( a b(a-b) \)[/tex]
- [tex]\( a b^2 + b^3 = b^2(a+b) \)[/tex]
- [tex]\( a^2(a-b) \)[/tex]

Total volume expression:
[tex]\[ a b(a-b) + b^2(a+b) + a^2(a-b) = (a+b)\left(a^2 - a b + b^2\right) \][/tex]

Label the sections:
- Red (bottom)
- Blue (top back)
- Green (top front and top small piece)



Answer :

Sure, let’s write a detailed, step-by-step solution to determine the total volume of the rectangular prisms and express it in a simplified form.

### Step-by-Step Solution

1. Identify Dimensions and Volumes of Each Prism:

- Red Prism (Bottom):
- Height: [tex]\(a\)[/tex]
- Length: [tex]\(b\)[/tex]
- Width: [tex]\(a - b\)[/tex]
- Volume: [tex]\(a \cdot b \cdot (a - b) = ab(a - b)\)[/tex]

- Blue Prism (Top Back):
- Height: [tex]\(b\)[/tex]
- Length: [tex]\(b\)[/tex]
- Width: [tex]\(a + b\)[/tex]
- Volume: [tex]\(b \cdot b \cdot (a + b) = b^2(a + b)\)[/tex]

- Green Prisms (Top Front and Top Small Piece):
- Height: [tex]\(a\)[/tex]
- Length: [tex]\(a\)[/tex]
- Width: [tex]\(a - b\)[/tex]
- Volume: [tex]\(a \cdot a \cdot (a - b) = a^2(a - b)\)[/tex]

2. Total Volume Calculation:
We sum the volumes of each individual prism to get the total volume of the combined structure.

[tex]\[ \text{Total Volume} = \text{Volume of Red Prism} + \text{Volume of Blue Prism} + \text{Volume of Green Prisms} \][/tex]

Substitute the expressions for each volume:

[tex]\[ \text{Total Volume} = ab(a - b) + b^2(a + b) + a^2(a - b) \][/tex]

3. Simplify the Expression:
Combine the terms and simplify the expression.

[tex]\[ \text{Total Volume} = ab(a - b) + b^2(a + b) + a^2(a - b) \][/tex]

Distribute each term:

[tex]\[ ab(a - b) = a^2b - ab^2 \][/tex]

[tex]\[ b^2(a + b) = ab^2 + b^3 \][/tex]

[tex]\[ a^2(a - b) = a^3 - a^2b \][/tex]

Combine all the terms:

[tex]\[ \text{Total Volume} = (a^2b - ab^2) + (ab^2 + b^3) + (a^3 - a^2b) \][/tex]

Observe that some terms cancel each other out:

[tex]\[ a^2b - a^2b - ab^2 + ab^2 + b^3 + a^3 \][/tex]

This simplifies to:

[tex]\[ \text{Total Volume} = a^3 + b^3 \][/tex]

4. Express the simplified form:
Notice that [tex]\(a^3 + b^3\)[/tex] can also be factored as:

[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]

Therefore, the simplified expression for the total volume of the combined prisms is:

[tex]\[ \boxed{(a + b)(a^2 - ab + b^2)} \][/tex]