To solve the expression [tex]\(\sqrt[3]{x^2}\)[/tex], follow these steps:
1. Identify the expression inside the radical:
The expression inside the radical is [tex]\(x^2\)[/tex].
2. Understand the meaning of the cube root:
The cube root of a number [tex]\(a\)[/tex] is a value that, when multiplied by itself three times, gives [tex]\(a\)[/tex]. For instance, [tex]\(\sqrt[3]{a}\)[/tex] can be written as [tex]\(a^{1/3}\)[/tex].
3. Apply the cube root to the given expression:
We need to find the cube root of [tex]\(x^2\)[/tex]. This can be written as:
[tex]\[
\sqrt[3]{x^2}
\][/tex]
4. Express the cube root in terms of exponents:
The cube root of [tex]\(x^2\)[/tex] is equivalent to raising [tex]\(x^2\)[/tex] to the power of [tex]\(1/3\)[/tex]:
[tex]\[
\sqrt[3]{x^2} = (x^2)^{1/3}
\][/tex]
5. Simplify the exponent:
When you raise a power to another power, you multiply the exponents. Therefore, [tex]\( (x^2)^{1/3} \)[/tex] simplifies to:
[tex]\[
(x^2)^{1/3} = x^{2 \cdot (1/3)}
\][/tex]
6. Perform the multiplication of the exponents:
Multiply the exponents 2 and [tex]\(1/3\)[/tex]:
[tex]\[
2 \cdot \frac{1}{3} = \frac{2}{3}
\][/tex]
Thus,
[tex]\[
x^{2 \cdot (1/3)} = x^{2/3}
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt[3]{x^2}\)[/tex] is:
[tex]\[
\sqrt[3]{x^2} = x^{2/3}
\][/tex]
