Questions: Perform the indicated operation. [ (m-10 n)^3(m+10 n) ] [ (m-10 n)^3(m+10 n)= ] (Simplify your answer. Use integers or fractions for any numbers in

Solution Steps

We start with the expression \((m - 10n)^{3}(m + 10n)\). First, we need to expand the cubic term \((m - 10n)^{3}\).

Using the binomial expansion formula, we have: \[ (m - 10n)^{3} = m^{3} - 3 \cdot m^{2} \cdot (10n) + 3 \cdot m \cdot (10n)^{2} - (10n)^{3} \] This simplifies to: \[ m^{3} - 30mn + 300n^{2} - 1000n^{3} \]

Next, we multiply the expanded cubic term by the linear term \((m + 10n)\): \[ (m^{3} - 30mn + 300n^{2} - 1000n^{3})(m + 10n) \]

We distribute each term in the cubic expression by each term in the linear expression: \[ = m^{3} \cdot m + m^{3} \cdot 10n - 30mn \cdot m - 30mn \cdot 10n + 300n^{2} \cdot m + 300n^{2} \cdot 10n - 1000n^{3} \cdot m - 1000n^{3} \cdot 10n \]

This results in: \[ = m^{4} + 10m^{3}n - 30m^{2}n - 300n^{2}m + 300mn^{2} + 3000n^{3} - 1000mn^{3} - 10000n^{4} \]

Now, we collect and simplify the like terms: \[ = m^{4} - 20m^{3}n + 2000mn^{3} - 10000n^{4} \]

Thus, the fully expanded expression is: \[ (m - 10n)^{3}(m + 10n) = m^{4} - 20m^{3}n + 2000mn^{3} - 10000n^{4} \]

Final Answer