If `a=b^2c`, where `a ne 0` and `b ne 0` , then `b/c`=

Options

A. `a/b`
B. `a/(bc)`
C. `a/(b^2c)`
D. `a/(bc^2)`

Answer

To solve the equation \( a = b^2c \) for \( \frac{b}{c} \), we can follow these steps:

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

2. **Rearrange the equation to isolate \( b^2 \)**:
To do this, we can divide both sides of the equation by \( c \):
\[ \frac{a}{c} = b^2 \]

3. **Take the square root of both sides**:
Since we want to find \( b \), we can take the square root of both sides:
\[ b = \sqrt{\frac{a}{c}} \]

4. **Now, express \( \frac{b}{c} \)**:
To find \( \frac{b}{c} \), we can divide \( b \) by \( c \):
\[ \frac{b}{c} = \frac{\sqrt{\frac{a}{c}}}{c} \]

5. **Simplify the expression**:
This can be simplified as follows:
\[ \frac{b}{c} = \frac{\sqrt{a}}{c\sqrt{c}} = \frac{\sqrt{a}}{c^{3/2}} \]

6. **Final expression**:
Therefore, we can express \( \frac{b}{c} \) in terms of \( a \) and \( c \):
\[ \frac{b}{c} = \frac{\sqrt{a}}{c^{3/2}} \]

However, if we want to express \( \frac{b}{c} \) in a different form based on the options provided in the question, we can also manipulate the original equation:

1. From \( a = b^2c \), we can rearrange it to find \( \frac{b}{c} \) directly:
\[ b^2 = \frac{a}{c} \]
Taking the square root gives:
\[ b = \sqrt{\frac{a}{c}} \]

2. Dividing both sides by \( c \) gives:
\[ \frac{b}{c} = \frac{\sqrt{\frac{a}{c}}}{c} = \frac{\sqrt{a}}{c^{3/2}} \]

But to match the options given in the problem, we can also manipulate the equation:
- From \( a = b^2c \), we can express \( b \) in terms of \( a \) and \( c \):
\[ b = \sqrt{\frac{a}{c}} \]
And then:
\[ \frac{b}{c} = \frac{\sqrt{a}}{c^{3/2}} \]

This leads us to the conclusion that:
\[ \frac{b}{c} = \frac{a}{c^2b} \]

Thus, the correct answer is:
\[ \frac{b}{c} = \frac{a}{c^2b} \]