To solve the expression [tex]\(\sqrt[3]{125 x^6}\)[/tex], follow these steps:
1. Understand the Third Root: The cube root of a number [tex]\(a\)[/tex] is a value that, when multiplied by itself three times, gives the number [tex]\(a\)[/tex]. For example, [tex]\(\sqrt[3]{a} = a^{1/3}\)[/tex].
2. Break Down the Expression:
The expression we need to simplify is [tex]\(\sqrt[3]{125 x^6}\)[/tex].
3. Separate the Constant and the Variable:
Notice that the expression inside the cube root can be separated into two parts: the constant (125) and the variable part ([tex]\(x^6\)[/tex]). So, we have:
[tex]\[
\sqrt[3]{125 x^6} = \sqrt[3]{125} \cdot \sqrt[3]{x^6}
\][/tex]
4. Simplify the Constant Part:
The cube root of 125 can be simplified as follows:
[tex]\[
\sqrt[3]{125} = \sqrt[3]{5^3} = 5
\][/tex]
5. Simplify the Variable Part:
Now, consider the variable part [tex]\(\sqrt[3]{x^6}\)[/tex]. When taking the cube root of [tex]\(x^6\)[/tex], you can use the rule that [tex]\((x^a)^{1/b} = x^{a/b}\)[/tex] to get:
[tex]\[
\sqrt[3]{x^6} = (x^6)^{1/3} = x^{6 \cdot \frac{1}{3}} = x^2
\][/tex]
6. Combine the Results:
Multiplying the simplified parts together, we get:
[tex]\[
\sqrt[3]{125 x^6} = 5 \cdot x^2
\][/tex]
So the simplified expression is:
[tex]\[
5 \cdot x^2
\][/tex]
Thus, [tex]\(\sqrt[3]{125 x^6} = 5 x^2\)[/tex].
