7. Simplify each of the following expressions. Assume that \( b \neq 0 \). Verify your \( \begin{array}{ll}\text { a. } m^{4}+(2 m)^{4} & \text { d. } \frac{a b^{3}}{b}+(a b)^{2}-a b^{2} \\ \text { b. } w^{8}\left(w^{2}+1\right) & \text { e. } a(-2 b)^{3} \\ \text { c. } k^{3} k^{5}-4\left(k^{4}\right)^{2} & \text { f. } M P^{2}\left(M P^{4}+1\right)\end{array} \)

Simplify the expression by following steps: - step0: Solution: \(m^{4}+\left(2m\right)^{4}\) - step1: Rewrite the expression: \(m^{4}+16m^{4}\) - step2: Collect like terms: \(\left(1+16\right)m^{4}\) - step3: Add the numbers: \(17m^{4}\) Calculate or simplify the expression \( M P^{2}(M P^{4}+1) \). Simplify the expression by following steps: - step0: Solution: \(MP^{2}\left(MP^{4}+1\right)\) - step1: Apply the distributive property: \(MP^{2}MP^{4}+MP^{2}\times 1\) - step2: Multiply the terms: \(M^{2}P^{6}+MP^{2}\) Calculate or simplify the expression \( k^{3} k^{5}-4(k^{4})^{2} \). Simplify the expression by following steps: - step0: Solution: \(k^{3}\times k^{5}-4\left(k^{4}\right)^{2}\) - step1: Multiply the exponents: \(k^{3}\times k^{5}-4k^{4\times 2}\) - step2: Multiply the numbers: \(k^{3}\times k^{5}-4k^{8}\) - step3: Multiply the terms: \(k^{8}-4k^{8}\) - step4: Collect like terms: \(\left(1-4\right)k^{8}\) - step5: Subtract the numbers: \(-3k^{8}\) Calculate or simplify the expression \( w^{8}(w^{2}+1) \). Simplify the expression by following steps: - step0: Solution: \(w^{8}\left(w^{2}+1\right)\) - step1: Apply the distributive property: \(w^{8}\times w^{2}+w^{8}\times 1\) - step2: Multiply the terms: \(w^{10}+w^{8}\) Calculate or simplify the expression \( \frac{a b^{3}}{b}+(a b)^{2}-a b^{2} \). Simplify the expression by following steps: - step0: Solution: \(\frac{ab^{3}}{b}+\left(ab\right)^{2}-ab^{2}\) - step1: Divide the terms: \(ab^{2}+\left(ab\right)^{2}-ab^{2}\) - step2: Rewrite the expression: \(ab^{2}+a^{2}b^{2}-ab^{2}\) - step3: Collect like terms: \(a^{2}b^{2}\) Calculate or simplify the expression \( a(-2 b)^{3} \). Simplify the expression by following steps: - step0: Solution: \(a\left(-2b\right)^{3}\) - step1: Rewrite the expression: \(a\left(-8b^{3}\right)\) - step2: Calculate: \(-8ab^{3}\) Here are the simplified expressions for each part of the problem: ### a. \( m^{4}+(2 m)^{4} \) 1. Calculate \( (2m)^{4} = 16m^{4} \). 2. Combine: \( m^{4} + 16m^{4} = 17m^{4} \). **Result:** \( 17m^{4} \) --- ### b. \( w^{8}(w^{2}+1) \) 1. Distribute \( w^{8} \) into \( (w^{2}+1) \): - \( w^{8} \cdot w^{2} = w^{10} \) - \( w^{8} \cdot 1 = w^{8} \) 2. Combine: \( w^{10} + w^{8} \). **Result:** \( w^{10} + w^{8} \) --- ### c. \( k^{3} k^{5}-4(k^{4})^{2} \) 1. Calculate \( k^{3} k^{5} = k^{8} \). 2. Calculate \( (k^{4})^{2} = k^{8} \) and then \( -4(k^{4})^{2} = -4k^{8} \). 3. Combine: \( k^{8} - 4k^{8} = -3k^{8} \). **Result:** \( -3k^{8} \) --- ### d. \( \frac{a b^{3}}{b}+(a b)^{2}-a b^{2} \) 1. Simplify \( \frac{a b^{3}}{b} = ab^{2} \). 2. Calculate \( (ab)^{2} = a^{2}b^{2} \). 3. Combine: \( ab^{2} + a^{2}b^{2} - ab^{2} = a^{2}b^{2} \). **Result:** \( a^{2}b^{2} \) --- ### e. \( a(-2 b)^{3} \) 1. Calculate \( (-2b)^{3} = -8b^{3} \). 2. Multiply: \( a \cdot (-8b^{3}) = -8ab^{3} \). **Result:** \( -8ab^{3} \) --- ### f. \( M P^{2}(M P^{4}+1) \) 1. Distribute \( MP^{2} \): - \( MP^{2} \cdot MP^{4} = M^{2}P^{6} \) - \( MP^{2} \cdot 1 = MP^{2} \) 2. Combine: \( M^{2}P^{6} + MP^{2} \). **Result:** \( M^{2}P^{6} + MP^{2} \) --- These are the simplified forms of each expression.