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

Question

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

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Solution:
  $$m^{4}+\left(2might)^{4}$$
- step1: Rewrite the expression:
  $$m^{4}+16m^{4}$$
- step2: Collect like terms:
  $$\left(1+16ight)m^{4}$$
- 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:
  $$MP^{2}\left(MP^{4}+1ight)$$
- step1: Apply the distributive property:
  $$MP^{2}MP^{4}+MP^{2}	imes 1$$
- 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:
  $$k^{3}	imes k^{5}-4\left(k^{4}ight)^{2}$$
- step1: Multiply the exponents:
  $$k^{3}	imes k^{5}-4k^{4	imes 2}$$
- step2: Multiply the numbers:
  $$k^{3}	imes k^{5}-4k^{8}$$
- step3: Multiply the terms:
  $$k^{8}-4k^{8}$$
- step4: Collect like terms:
  $$\left(1-4ight)k^{8}$$
- step5: Subtract the numbers:
  $$-3k^{8}$$
Calculate or simplify the expression $$ w^{8}(w^{2}+1) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$w^{8}\left(w^{2}+1ight)$$
- step1: Apply the distributive property:
  $$w^{8}	imes w^{2}+w^{8}	imes 1$$
- 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:
  $$\frac{ab^{3}}{b}+\left(abight)^{2}-ab^{2}$$
- step1: Divide the terms:
  $$ab^{2}+\left(abight)^{2}-ab^{2}$$
- 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:
  $$a\left(-2bight)^{3}$$
- step1: Rewrite the expression:
  $$a\left(-8b^{3}ight)$$
- 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.
Here are the simplified expressions: a. $$ 17m^{4} $$ b. $$ w^{10} + w^{8} $$ c. $$ -3k^{8} $$ d. $$ a^{2}b^{2} $$ e. $$ -8ab^{3} $$ f. $$ M^{2}P^{6} + MP^{2} $$

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Algebra Questions