Answer:
B. [tex]18x^{2} y^{2} \sqrt[3]{3xy^{2} }[/tex]
Step-by-step explanation:
To simplify this expression, use the fact that the root of a number (in this case is the cube root) can be expressed like a fractional exponent (1/3). Using this, the expression changes to:
[tex]3x(648x^{4}y^{8})^{(1/3)}[/tex]
Next step is to put the exponent inside the parenthesis:
[tex]3x(648^{1/3}x^{4/3}y^{8/3})[/tex]
Find the prime factorization of 648:
648 =3⋅3⋅3⋅3⋅2⋅2⋅2
648=3⁴∗2³
[tex]3x(3^{(4/3)}2^{(3/3)}x^{4/3}y^{8/3})[/tex]
Change all improper fractions in exponent to mixed fractions
[tex]3x(3^{1(1/3)}2^{1}x^{1(1/3)}y^{2(2/3)})[/tex]
Separate integers exponents from fractional:
[tex]3x(3\cdot3^{(1/3)}\cdot 2 \cdot x\cdot x^{1/3}\cdot y^{2}\cdot y^{2/3})[/tex]
Re-arrange (all numbers with fractional exponents must be together):
[tex]3x(3 \cdot 2\cdot x\cdot y^{2}\cdot 3^{(1/3)}x^{1/3}y^{2/3})[/tex]
Multiply the 3x with the numbers that have an integer exponent:
[tex]18x^{2}y^{2}(3^{(1/3)}x^{1/3}y^{2/3})[/tex]
Take out the exponent 1/3 from the parenthesis:
[tex]18x^{2}y^{2}(3xy^{2})^{1/3}[/tex]
And change the representation of the root to use a radical symbol
[tex]18x^{2} y^{2} \sqrt[3]{3xy^{2} }[/tex]
