Simplify the radicals in the given expression:

[tex]\[ 8 \sqrt[3]{a^4 b^3 c^2} - 14 a b \sqrt[3]{a c^2} \][/tex]

A. [tex]\( 8 a b \sqrt[3]{a c^2} - 14 a b \sqrt[3]{a c^2} \)[/tex]

B. [tex]\( 8 a^2 b c \sqrt[3]{b} - 14 a b c \sqrt[3]{a} \)[/tex]

C. [tex]\( 8 a b \sqrt{a c^2} - 14 a b \sqrt[3]{a c^2} \)[/tex]

D. [tex]\( 8 a^2 b c \sqrt{b} - 14 a b c \sqrt{a} \)[/tex]



Answer :

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]