Answer
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. \( M^{2}P^{6} + MP^{2} \)
Solution
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.
Answered by UpStudy AI and reviewed by a Professional Tutor

Explain

Simplify this solution