Questions: ((-2 m^(-2) n^3 p^2)/(4 m^(-3) n^(-2) p^(-1)))^(-3)

Solution Steps

To simplify the given expression, we need to follow these steps:

1. Simplify the fraction inside the parentheses by dividing the coefficients and subtracting the exponents of like bases.
2. Apply the negative exponent to the simplified fraction.
3. Simplify the resulting expression.

We start with the expression

\[ \left(\frac{-2 m^{-2} n^{3} p^{2}}{4 m^{-3} n^{-2} p^{-1}}\right)^{-3} \]

First, we simplify the fraction inside the parentheses. The coefficients give us

\[ \frac{-2}{4} = -\frac{1}{2} \]

Next, we simplify the variables by applying the laws of exponents:

\[ m^{-2} \div m^{-3} = m^{-2 - (-3)} = m^{1} \]

\[ n^{3} \div n^{-2} = n^{3 - (-2)} = n^{5} \]

\[ p^{2} \div p^{-1} = p^{2 - (-1)} = p^{3} \]

Thus, the simplified fraction becomes

\[ -\frac{1}{2} m^{1} n^{5} p^{3} = -\frac{m n^{5} p^{3}}{2} \]

Now we apply the negative exponent of \(-3\):

\[ \left(-\frac{m n^{5} p^{3}}{2}\right)^{-3} = \left(-1\right)^{-3} \cdot \left(\frac{m n^{5} p^{3}}{2}\right)^{-3} \]

Calculating \((-1)^{-3}\) gives us \(-1\). For the fraction, we have:

\[ \left(\frac{1}{2}\right)^{-3} = 2^{3} = 8 \]

And for the variables:

\[ (m n^{5} p^{3})^{-3} = m^{-3} n^{-15} p^{-9} \]

Combining these results, we get:

\[ -1 \cdot 8 \cdot m^{-3} n^{-15} p^{-9} = -\frac{8}{m^{3} n^{15} p^{9}} \]

Final Answer