Which of the following equations expresses c in terms of a for all real numbers a, b and c such that `a^(3) = b` and `b^(2) = c` ?

Options

A. `c = a^(6)`
B. `c = a^(5)`
C. `c = 2a^(3)`
D. `c = 1/2 a`

Answer

To express \( c \) in terms of \( a \) given the equations \( a^3 = b \) and \( b^2 = c \), we can follow these steps:

### Step-by-Step Solution:

1. **Start with the given equations:**
\[ a^3 = b \]
\[ b^2 = c \]

2. **Substitute \( b \) in the second equation:**
Since we know \( b = a^3 \), we can substitute this into the equation for \( c \):
\[ c = b^2 = (a^3)^2 \]

3. **Apply the exponent rule:**
When raising a power to another power, we multiply the exponents:
\[ (a^3)^2 = a^{3 \times 2} = a^6 \]

4. **Final expression for \( c \):**
Therefore, we have:
\[ c = a^6 \]

### Conclusion:
The equation that expresses \( c \) in terms of \( a \) is:
\[ c = a^6 \]