Step-by-Step Solution:
We need to simplify the following expressions by combining like terms and simplifying any radicals:
#### Expression 1:
[tex]\[ 8 \sqrt[3]{a^4 b^3 c^2} - 14 a b \sqrt[3]{a c^2} \][/tex]
1. Identify the cube roots:
[tex]\[ 8 \sqrt[3]{a^4 b^3 c^2} \][/tex]
[tex]\[ 14 a b \sqrt[3]{a c^2} \][/tex]
2. Simplify each cube root individually:
- [tex]\( \sqrt[3]{a^4 b^3 c^2} \)[/tex] can be written as:
[tex]\[ \sqrt[3]{a^4 b^3 c^2} = a \cdot \sqrt[3]{a \cdot b^3 \cdot c^2} = a b \sqrt[3]{a c^2} \][/tex]
- Thus,
[tex]\[ 8 \sqrt[3]{a^4 b^3 c^2} = 8 a b \sqrt[3]{a c^2} \][/tex]
- Similarly,
[tex]\[ 14 a b \sqrt[3]{a c^2} = 14 a b \sqrt[3]{a c^2} \][/tex]
3. Combine like terms:
[tex]\[ 8 a b \sqrt[3]{a c^2} - 14 a b \sqrt[3]{a c^2} \][/tex]
4. Factor out the common term [tex]\( a b \sqrt[3]{a c^2} \)[/tex]:
[tex]\[ (8 - 14) a b \sqrt[3]{a c^2} \][/tex]
[tex]\[ -6 a b \sqrt[3]{a c^2} \][/tex]
#### Expression 2:
[tex]\[ 8 a b \sqrt[3]{a c^2} - 14 a b \sqrt[3]{a c^2} \][/tex]
1. Identify the cube roots:
[tex]\[ 8 a b \sqrt[3]{a c^2} \][/tex]
[tex]\[ 14 a b \sqrt[3]{a c^2} \][/tex]
2. Combine like terms directly:
[tex]\[ 8 a b \sqrt[3]{a c^2} - 14 a b \sqrt[3]{a c^2} \][/tex]
3. Factor out the common term [tex]\( a b \sqrt[3]{a c^2} \)[/tex]:
[tex]\[ (8 - 14) a b \sqrt[3]{a c^2} \][/tex]
[tex]\[ -6 a b \sqrt[3]{a c^2} \][/tex]
#### Expression 3:
[tex]\[ 8 a^2 b c \sqrt[3]{b} - 14 a b c \sqrt[3]{a} \][/tex]
1. Both terms have the common factor [tex]\( a b c \)[/tex]:
[tex]\[ 8 a^2 b c \sqrt[3]{b} = 8 a b c \cdot a \sqrt[3]{b} \][/tex]
[tex]\[ 14 a b c \sqrt[3]{a} = 14 a b c \sqrt[3]{a} \][/tex]
2. Factor out the common term [tex]\( a b c \)[/tex]:
[tex]\[ 8 a b c \cdot (a \sqrt[3]{b}) - 14 a b c \sqrt[3]{a} \][/tex]
3. Combine terms:
[tex]\[ a b c (8 a \sqrt[3]{b} - 14 \sqrt[3]{a}) \][/tex]
#### Expression 4:
[tex]\[ 8 a^2 b c \sqrt[3]{b} - 14 a b c \sqrt[3]{a} \][/tex]
This expression is the same as Expression 3, so the simplified form is:
[tex]\[ a b c (8 a \sqrt[3]{b} - 14 \sqrt[3]{a}) \][/tex]
#### Expression 5:
[tex]\[ 8 a^2 b c \sqrt{b} - 14 a b c \sqrt{a} \][/tex]
1. Both terms have the common factor [tex]\( a b c \)[/tex]:
[tex]\[ 8 a^2 b c \sqrt{b} = 8 a b c \cdot a \sqrt{b} \][/tex]
[tex]\[ 14 a b c \sqrt{a} = 14 a b c \sqrt{a} \][/tex]
2. Factor out the common term [tex]\( a b c \)[/tex]:
[tex]\[ 8 a b c \cdot (a \sqrt{b}) - 14 a b c \sqrt{a} \][/tex]
3. Combine terms:
[tex]\[ a b c (8 a \sqrt{b} - 14 \sqrt{a}) \][/tex]
#### Final Simplified Expressions:
1. [tex]\( \boxed{-6 a b \sqrt[3]{a c^2}} \)[/tex]
2. [tex]\( \boxed{-6 a b \sqrt[3]{a c^2}} \)[/tex]
3. [tex]\( \boxed{a b c (8 a \sqrt[3]{b} - 14 \sqrt[3]{a})} \)[/tex]
4. [tex]\( \boxed{a b c (8 a \sqrt[3]{b} - 14 \sqrt[3]{a})} \)[/tex]
5. [tex]\( \boxed{a b c (8 a \sqrt{b} - 14 \sqrt{a})} \)[/tex]
