Question

13 Which expression is equivalent to (45m ^ - 6 * rho ^ 2 * v ^ 12)/(15m ^ - 2 * rho ^ 8 * v ^ - 4) for all values of m, p, and v where expression is defined?
Q
B (3v ^ 16)/(m ^ 4 * p ^ 6)
A (3v ^ 8)/(m ^ 8 * p ^ 6)
C (30m ^ 3)/(p ^ 4 * v ^ 3)
D 30v ^ 3 m^ 3 p^ -

Answer 1

To simplify the expression \((45m^{-6} \rho^2 v^{12})/(15m^{-2} \rho^8 v^{-4})\), we will first simplify the coefficients, the terms involving \(m\), the terms involving \(\rho\), and the terms involving \(v\).

### Step 1: Simplify the Coefficients
\[
\frac{45}{15} = 3
\]

### Step 2: Simplify the \(m\) terms
\[
\frac{m^{-6}}{m^{-2}} = m^{-6 - (-2)} = m^{-6 + 2} = m^{-4}
\]

### Step 3: Simplify the \(\rho\) terms
\[
\frac{\rho^2}{\rho^8} = \rho^{2 - 8} = \rho^{-6}
\]

### Step 4: Simplify the \(v\) terms
\[
\frac{v^{12}}{v^{-4}} = v^{12 - (-4)} = v^{12 + 4} = v^{16}
\]

### Final Combined Expression
Putting it all together, we have:
\[
3 \cdot (m^{-4}) \cdot (\rho^{-6}) \cdot (v^{16}) = \frac{3v^{16}}{m^{4}\rho^{6}}
\]

### Choose the Equivalent Expression
The simplified expression \(\frac{3v^{16}}{m^{4} \rho^{6}}\) matches with option **B**:
\[
\text{B: } \frac{3v^{16}}{m^{4} \rho^{6}}
\]

Thus, the expression equivalent to \((45m^{-6} \rho^{2} v^{12})/(15m^{-2} \rho^{8} v^{-4})\) is option **B**.