Answer :
Let's solve the problem step by step to find the cube root of the expression [tex]\(8x^{27}\)[/tex].
### Step 1: Understand Cube Roots
The cube root of a product [tex]\((8 \times x^{27})\)[/tex] can be found individually for each part: the constant [tex]\(8\)[/tex] and the [tex]\(x^{27}\)[/tex].
### Step 2: Cube Root of the Constant
- Begin with the constant [tex]\(8\)[/tex].
- [tex]\(8\)[/tex] can be expressed as [tex]\(2^3\)[/tex].
- The cube root of [tex]\(8\)[/tex] is [tex]\(\sqrt[3]{8} = 2\)[/tex].
### Step 3: Cube Root of the Power
- Now, handle the variable portion [tex]\(x^{27}\)[/tex].
- When taking the cube root of a power, you divide the exponent by [tex]\(3\)[/tex]:
[tex]\[
\sqrt[3]{x^{27}} = x^{27/3} = x^9.
\][/tex]
### Step 4: Combine Results
- Combine the results of the constant and the variable:
[tex]\[
\text{Cube root of } 8x^{27} = 2 \times x^9.
\][/tex]
This gives us the final result: [tex]\(2x^9\)[/tex].
### Conclusion
The cube root of [tex]\(8x^{27}\)[/tex] is [tex]\(2x^9\)[/tex], and when compared with the given options, it's clear that the correct choice is:
[tex]\[
2x^9
\][/tex]
### Step 1: Understand Cube Roots
The cube root of a product [tex]\((8 \times x^{27})\)[/tex] can be found individually for each part: the constant [tex]\(8\)[/tex] and the [tex]\(x^{27}\)[/tex].
### Step 2: Cube Root of the Constant
- Begin with the constant [tex]\(8\)[/tex].
- [tex]\(8\)[/tex] can be expressed as [tex]\(2^3\)[/tex].
- The cube root of [tex]\(8\)[/tex] is [tex]\(\sqrt[3]{8} = 2\)[/tex].
### Step 3: Cube Root of the Power
- Now, handle the variable portion [tex]\(x^{27}\)[/tex].
- When taking the cube root of a power, you divide the exponent by [tex]\(3\)[/tex]:
[tex]\[
\sqrt[3]{x^{27}} = x^{27/3} = x^9.
\][/tex]
### Step 4: Combine Results
- Combine the results of the constant and the variable:
[tex]\[
\text{Cube root of } 8x^{27} = 2 \times x^9.
\][/tex]
This gives us the final result: [tex]\(2x^9\)[/tex].
### Conclusion
The cube root of [tex]\(8x^{27}\)[/tex] is [tex]\(2x^9\)[/tex], and when compared with the given options, it's clear that the correct choice is:
[tex]\[
2x^9
\][/tex]
