To simplify the given expression [tex]\(\sqrt[3]{40 w^8}\)[/tex] into its radical form, follow these steps:
1. Expression Analysis:
We are given [tex]\(\sqrt[3]{40 w^8}\)[/tex].
2. Separate the Coefficients and Variables:
The expression can be broken down into the product of the coefficient and the variable part:
[tex]\[
\sqrt[3]{40 w^8} = \sqrt[3]{40} \cdot \sqrt[3]{w^8}
\][/tex]
3. Simplify the Variable Part:
We need to simplify [tex]\(\sqrt[3]{w^8}\)[/tex]:
[tex]\[
\sqrt[3]{w^8} = (w^8)^{1/3} = w^{8/3}
\][/tex]
4. Simplify the Coefficient:
Next, we simplify [tex]\(\sqrt[3]{40}\)[/tex]. Since 40 can be factored into prime numbers, we get:
[tex]\[
40 = 2^3 \times 5
\][/tex]
Thus,
[tex]\[
\sqrt[3]{40} = \sqrt[3]{2^3 \times 5}
\][/tex]
By the properties of radicals, this can be further simplified:
[tex]\[
\sqrt[3]{2^3 \times 5} = \sqrt[3]{2^3} \cdot \sqrt[3]{5} = 2 \cdot \sqrt[3]{5}
\][/tex]
5. Combine the Results:
Now, combining the simplified variable and the coefficient, we get:
[tex]\[
\sqrt[3]{40 w^8} = 2 \cdot \sqrt[3]{5} \cdot w^{8/3}
\][/tex]
So, the simplified form of [tex]\(\sqrt[3]{40 w^8}\)[/tex] is:
[tex]\[
2 \cdot 5^{1/3} \cdot w^{8/3}
\][/tex]
Thus, the final answer is:
[tex]\[
2 \cdot 5^{1/3} \cdot w^{8/3}
\][/tex]
