Find the forth proportional to `12 a, 9a,8a^(2)`

Options

A. `4a^(2)`
B. `6a^(2)`
C. `5a`
D. `7a^(2)`

Answer

To find the fourth proportional to the numbers \(12a\), \(9a\), and \(8a^2\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Fourth Proportional**:
The fourth proportional to three quantities \(a\), \(b\), and \(c\) is a number \(x\) such that the ratio \( \frac{a}{b} = \frac{c}{x} \).

2. **Set Up the Equation**:
Here, we have:
\[ \frac{12a}{9a} = \frac{8a^2}{x} \]

3. **Simplify the Left Side**:
We can simplify \( \frac{12a}{9a} \):
\[ \frac{12}{9} = \frac{4}{3} \]
So, our equation now looks like:
\[ \frac{4}{3} = \frac{8a^2}{x} \]

4. **Cross Multiply**:
Cross multiplying gives us:
\[ 4x = 3 \cdot 8a^2 \]

5. **Calculate the Right Side**:
Calculate \(3 \cdot 8a^2\):
\[ 3 \cdot 8a^2 = 24a^2 \]
So the equation becomes:
\[ 4x = 24a^2 \]

6. **Solve for \(x\)**:
Now, divide both sides by 4 to find \(x\):
\[ x = \frac{24a^2}{4} = 6a^2 \]

7. **Conclusion**:
Therefore, the fourth proportional to \(12a\), \(9a\), and \(8a^2\) is:
\[ x = 6a^2 \]

### Final Answer:
The fourth proportional is \(6a^2\).

---