To simplify the expression [tex]\(\sqrt[3]{27 x^3}\)[/tex], we can start by recognizing that this represents the cube root of a product.
Step-by-step simplification:
1. Identify the Cube Root:
The expression [tex]\(\sqrt[3]{27 x^3}\)[/tex] can be interpreted as applying the cube root to the factors inside the parentheses.
2. Break Down the Expression:
Break the cube root into the product of separate cube roots:
[tex]\[
\sqrt[3]{27 x^3} = \sqrt[3]{27} \cdot \sqrt[3]{x^3}
\][/tex]
3. Simplify Each Part:
- [tex]\(\sqrt[3]{27}\)[/tex]: Since 27 is [tex]\(3^3\)[/tex], the cube root of 27 is 3:
[tex]\[
\sqrt[3]{27} = 3
\][/tex]
- [tex]\(\sqrt[3]{x^3}\)[/tex]: The cube root of [tex]\(x^3\)[/tex] simplifies to [tex]\(x\)[/tex]:
[tex]\[
\sqrt[3]{x^3} = x
\][/tex]
4. Combine the Results:
Multiplying these results together:
[tex]\[
3 \cdot x = 3x
\][/tex]
Thus, the simplified form of the expression [tex]\(\sqrt[3]{27 x^3}\)[/tex] is [tex]\(3x\)[/tex].
So, the detailed solution step-by-step yields:
[tex]\[
\sqrt[3]{27 x^3} = 3x
\][/tex]
