College

What is the cube root of [tex]$27x^{18}$[/tex]?

A. [tex]$3x^6$[/tex]
B. [tex]$3x^9$[/tex]
C. [tex]$9x^6$[/tex]
D. [tex]$9x^9$[/tex]

Answer :

We start with the expression

[tex]$$
\sqrt[3]{27 x^{18}}.
$$[/tex]

This expression can be separated into the cube root of a product:

[tex]$$
\sqrt[3]{27 x^{18}} = \sqrt[3]{27} \cdot \sqrt[3]{x^{18}}.
$$[/tex]

First, notice that

[tex]$$
27 = 3^3,
$$[/tex]

so its cube root is

[tex]$$
\sqrt[3]{27} = 3.
$$[/tex]

Next, for the variable part, we use the property of exponents for radicals:

[tex]$$
\sqrt[3]{x^{18}} = x^{\frac{18}{3}} = x^6.
$$[/tex]

Thus, the entire expression simplifies to

[tex]$$
\sqrt[3]{27 x^{18}} = 3 \cdot x^6 = 3 x^6.
$$[/tex]

Therefore, the cube root of [tex]$27x^{18}$[/tex] is [tex]$\boxed{3 x^6}$[/tex].