If x varies as `m^(th)` power of y, y varies as `n^(th)` power of z and x varies as `p^(th)` power of z then which one of the following is true?

Options

A. `p=m+n`
B. `p=m-n`
C. `p=mn`
D. None of these

Answer

To solve the problem step by step, we will analyze the relationships given in the question:

1. **Understanding the Variations**:
- We know that \( x \) varies as the \( m^{th} \) power of \( y \). This can be expressed mathematically as:
\[ x = k_1 y^m \]
where \( k_1 \) is a constant of variation.

2. **Expressing \( y \) in terms of \( x \)**:
- Rearranging the equation gives us:
\[ y = \frac{x}{k_1^{1/m}} \]

3. **Next, we know that \( y \) varies as the \( n^{th} \) power of \( z \)**:
- This can be expressed as:
\[ y = k_2 z^n \]
where \( k_2 \) is another constant of variation.

4. **Expressing \( z \) in terms of \( y \)**:
- Rearranging this equation gives us:
\[ z = \frac{y}{k_2^{1/n}} \]

5. **Now substituting \( y \) from step 2 into this equation**:
- Substituting \( y = k_1 z^m \) into \( z = \frac{y}{k_2^{1/n}} \) gives:
\[ z = \frac{k_1 z^m}{k_2^{1/n}} \]

6. **Now we know that \( x \) varies as the \( p^{th} \) power of \( z \)**:
- This can be expressed as:
\[ x = k_3 z^p \]
where \( k_3 \) is another constant of variation.

7. **Expressing \( z \) in terms of \( x \)**:
- Rearranging gives us:
\[ z = \frac{x}{k_3^{1/p}} \]

8. **Substituting \( z \) back into the equation for \( y \)**:
- We can express \( y \) in terms of \( z \) and then substitute it back into the equation for \( x \):
\[ y = k_2 \left(\frac{x}{k_3^{1/p}}\right)^n \]

9. **Equating the two expressions for \( y \)**:
- From the two expressions for \( y \):
\[ \frac{x}{k_1^{1/m}} = k_2 \left(\frac{x}{k_3^{1/p}}\right)^n \]

10. **Simplifying the expressions**:
- This leads to:
\[ x^{1 - n} = k_2 k_3^{-n/p} k_1^{1/m} \]

11. **Final Relationship**:
- After simplifying, we find that:
\[ p = mn \]

Thus, the correct relationship is:
\[ p = mn \]