Let's simplify each expression to determine which one is in its simplest form.
### Expression A:
[tex]\[ 3 \sqrt{a^2 b^4} \][/tex]
First, use the property of square roots that [tex]\(\sqrt{xy} = \sqrt{x} \cdot \sqrt{y}\)[/tex]:
[tex]\[ 3 \sqrt{a^2 b^4} = 3 \sqrt{a^2} \cdot \sqrt{b^4} \][/tex]
Since [tex]\(\sqrt{a^2} = a\)[/tex] and [tex]\(\sqrt{b^4} = b^2\)[/tex]:
[tex]\[ 3 \sqrt{a^2 b^4} = 3 \cdot a \cdot b^2 = 3ab^2 \][/tex]
### Expression B:
[tex]\[ 3 a^3 \sqrt{3 b^4} \][/tex]
Again, use the property of square roots [tex]\(\sqrt{xy} = \sqrt{x} \cdot \sqrt{y}\)[/tex]:
[tex]\[ 3 a^3 \sqrt{3 b^4} = 3 a^3 \sqrt{3} \cdot \sqrt{b^4} \][/tex]
Since [tex]\(\sqrt{b^4} = b^2\)[/tex]:
[tex]\[ 3 a^3 \sqrt{3 b^4} = 3 a^3 \cdot \sqrt{3} \cdot b^2 = 3 a^3 b^2 \sqrt{3} \][/tex]
### Expression C:
[tex]\[ a b \sqrt{9 a b} \][/tex]
First, notice that [tex]\(9 = 3^2\)[/tex]:
[tex]\[ a b \sqrt{9 a b} = a b \sqrt{9} \cdot \sqrt{a b} \][/tex]
Since [tex]\(\sqrt{9} = 3\)[/tex]:
[tex]\[ a b \sqrt{9 a b} = a b \cdot 3 \cdot \sqrt{a b} = 3 a b \sqrt{a b} \][/tex]
### Expression D:
[tex]\[ 3 a b \sqrt{3 a b} \][/tex]
This expression does not break down further using properties of square roots:
[tex]\[ 3 a b \sqrt{3 a b} \][/tex]
### Conclusion:
Now, let's compare the simplified forms of the expressions:
- Expression A: [tex]\( 3 a b^2 \)[/tex]
- Expression B: [tex]\( 3 a^3 b^2 \sqrt{3} \)[/tex]
- Expression C: [tex]\( 3 a b \sqrt{a b} \)[/tex]
- Expression D: [tex]\( 3 a b \sqrt{3 a b} \)[/tex]
Expression A, [tex]\(3 a b^2\)[/tex], is the simplest form without any remaining square roots or higher powers.
Therefore, the correct answer is:
A. [tex]\( 3 \sqrt{a^2 b^4} \)[/tex]
