Answer :
Let us simplify the given expressions step-by-step.
### Expression 1
[tex]\[ 8 \sqrt[3]{a^4 b^3 c^2} - 14 a b \sqrt[3]{a c^2} \][/tex]
First, let's rewrite each term to make the cube roots more explicit:
1. [tex]\( 8 \sqrt[3]{a^4 b^3 c^2} = 8 \cdot (a^4 b^3 c^2)^{1/3} = 8 \cdot a^{4/3} b c^{2/3} \)[/tex]
2. [tex]\( 14 a b \sqrt[3]{a c^2} = 14 a b \cdot (a c^2)^{1/3} = 14 a b \cdot a^{1/3} c^{2/3} = 14 a^{4/3} b c^{2/3} \)[/tex]
So the expression becomes:
[tex]\[ 8 a^{4/3} b c^{2/3} - 14 a^{4/3} b c^{2/3} \][/tex]
Combine the like terms:
[tex]\[ (8 - 14) a^{4/3} b c^{2/3} = -6 a^{4/3} b c^{2/3} \][/tex]
### Expression 2
[tex]\[ 8 a b \sqrt[3]{a c^2} - 14 a b \sqrt[3]{a c^2} \][/tex]
These terms are like terms. Using the properties of cube roots:
1. [tex]\( 8 a b \sqrt[3]{a c^2} = 8 a b (a c^2)^{1/3} \)[/tex]
2. [tex]\( 14 a b \sqrt[3]{a c^2} = 14 a b (a c^2)^{1/3} \)[/tex]
So, once they are like terms:
[tex]\[ (8 - 14) a b (a c^2)^{1/3} = -6 a b (a c^2)^{1/3} \][/tex]
### Expression 3
[tex]\[ 8 a^2 b c \sqrt[3]{b} - 14 a b c \sqrt[3]{a} \][/tex]
Rewrite these terms to make the cube roots more explicit:
1. [tex]\( 8 a^2 b c \sqrt[3]{b} = 8 a^2 b c b^{1/3} = 8 a^2 b^{4/3} c \)[/tex]
2. [tex]\( 14 a b c \sqrt[3]{a} = 14 a b c a^{1/3} = 14 a^{4/3} b c \)[/tex]
So the expression does not combine directly:
[tex]\[ 8 a^2 b^{4/3} c - 14 a^{4/3} b c \][/tex]
### Expression 4
[tex]\[ 8 a b \sqrt[3]{a c^2} - 14 a b \sqrt[3]{a c^2} \][/tex]
These terms are the same as in Expression 2:
1. [tex]\( 8 a b \sqrt[3]{a c^2} = 8 a b (a c^2)^{1/3} \)[/tex]
2. [tex]\( 14 a b \sqrt[3]{a c^2} = 14 a b (a c^2)^{1/3} \)[/tex]
So:
[tex]\[ (8 - 14) a b (a c^2)^{1/3} = -6 a b (a c^2)^{1/3} \][/tex]
### Expression 5
[tex]\[ 8 a^2 b c \sqrt{b} - 14 a b c \sqrt{a} \][/tex]
Rewrite these terms:
1. [tex]\( 8 a^2 b c \sqrt{b} = 8 a^2 b c b^{1/2} = 8 a^2 b^{3/2} c \)[/tex]
2. [tex]\( 14 a b c \sqrt{a} = 14 a b c a^{1/2} = 14 a^{3/2} b c \)[/tex]
So the expression does not combine directly:
[tex]\[ 8 a^2 b^{3/2} c - 14 a^{3/2} b c \][/tex]
To summarize, here are the simplified results for each expression:
1. [tex]\(-6 a^{4/3} b c^{2/3}\)[/tex]
2. [tex]\(-6 a b (a c^2)^{1/3}\)[/tex]
3. [tex]\(8 a^2 b^{4/3} c - 14 a^{4/3} b c\)[/tex]
4. [tex]\(-6 a b (a c^2)^{1/3}\)[/tex]
5. [tex]\(8 a^2 b^{3/2} c - 14 a^{3/2} b c\)[/tex]
These are the simplified forms of the radicals in each of the given expressions.
### Expression 1
[tex]\[ 8 \sqrt[3]{a^4 b^3 c^2} - 14 a b \sqrt[3]{a c^2} \][/tex]
First, let's rewrite each term to make the cube roots more explicit:
1. [tex]\( 8 \sqrt[3]{a^4 b^3 c^2} = 8 \cdot (a^4 b^3 c^2)^{1/3} = 8 \cdot a^{4/3} b c^{2/3} \)[/tex]
2. [tex]\( 14 a b \sqrt[3]{a c^2} = 14 a b \cdot (a c^2)^{1/3} = 14 a b \cdot a^{1/3} c^{2/3} = 14 a^{4/3} b c^{2/3} \)[/tex]
So the expression becomes:
[tex]\[ 8 a^{4/3} b c^{2/3} - 14 a^{4/3} b c^{2/3} \][/tex]
Combine the like terms:
[tex]\[ (8 - 14) a^{4/3} b c^{2/3} = -6 a^{4/3} b c^{2/3} \][/tex]
### Expression 2
[tex]\[ 8 a b \sqrt[3]{a c^2} - 14 a b \sqrt[3]{a c^2} \][/tex]
These terms are like terms. Using the properties of cube roots:
1. [tex]\( 8 a b \sqrt[3]{a c^2} = 8 a b (a c^2)^{1/3} \)[/tex]
2. [tex]\( 14 a b \sqrt[3]{a c^2} = 14 a b (a c^2)^{1/3} \)[/tex]
So, once they are like terms:
[tex]\[ (8 - 14) a b (a c^2)^{1/3} = -6 a b (a c^2)^{1/3} \][/tex]
### Expression 3
[tex]\[ 8 a^2 b c \sqrt[3]{b} - 14 a b c \sqrt[3]{a} \][/tex]
Rewrite these terms to make the cube roots more explicit:
1. [tex]\( 8 a^2 b c \sqrt[3]{b} = 8 a^2 b c b^{1/3} = 8 a^2 b^{4/3} c \)[/tex]
2. [tex]\( 14 a b c \sqrt[3]{a} = 14 a b c a^{1/3} = 14 a^{4/3} b c \)[/tex]
So the expression does not combine directly:
[tex]\[ 8 a^2 b^{4/3} c - 14 a^{4/3} b c \][/tex]
### Expression 4
[tex]\[ 8 a b \sqrt[3]{a c^2} - 14 a b \sqrt[3]{a c^2} \][/tex]
These terms are the same as in Expression 2:
1. [tex]\( 8 a b \sqrt[3]{a c^2} = 8 a b (a c^2)^{1/3} \)[/tex]
2. [tex]\( 14 a b \sqrt[3]{a c^2} = 14 a b (a c^2)^{1/3} \)[/tex]
So:
[tex]\[ (8 - 14) a b (a c^2)^{1/3} = -6 a b (a c^2)^{1/3} \][/tex]
### Expression 5
[tex]\[ 8 a^2 b c \sqrt{b} - 14 a b c \sqrt{a} \][/tex]
Rewrite these terms:
1. [tex]\( 8 a^2 b c \sqrt{b} = 8 a^2 b c b^{1/2} = 8 a^2 b^{3/2} c \)[/tex]
2. [tex]\( 14 a b c \sqrt{a} = 14 a b c a^{1/2} = 14 a^{3/2} b c \)[/tex]
So the expression does not combine directly:
[tex]\[ 8 a^2 b^{3/2} c - 14 a^{3/2} b c \][/tex]
To summarize, here are the simplified results for each expression:
1. [tex]\(-6 a^{4/3} b c^{2/3}\)[/tex]
2. [tex]\(-6 a b (a c^2)^{1/3}\)[/tex]
3. [tex]\(8 a^2 b^{4/3} c - 14 a^{4/3} b c\)[/tex]
4. [tex]\(-6 a b (a c^2)^{1/3}\)[/tex]
5. [tex]\(8 a^2 b^{3/2} c - 14 a^{3/2} b c\)[/tex]
These are the simplified forms of the radicals in each of the given expressions.
