To simplify the expression [tex]\( \sqrt[3]{40 w^8} \)[/tex], follow these steps:
1. Factorize the constant inside the cube root:
[tex]\[
40 = 8 \times 5
\][/tex]
Therefore, we rewrite the expression as:
[tex]\[
\sqrt[3]{40 w^8} = \sqrt[3]{8 \times 5 \times w^8}
\][/tex]
2. Utilize the properties of cube roots:
[tex]\[
\sqrt[3]{a \times b \times c} = \sqrt[3]{a} \times \sqrt[3]{b} \times \sqrt[3]{c}
\][/tex]
Applying this property to our expression, we get:
[tex]\[
\sqrt[3]{8 \times 5 \times w^8} = \sqrt[3]{8} \times \sqrt[3]{5} \times \sqrt[3]{w^8}
\][/tex]
3. Simplify each part individually:
- The cube root of 8:
[tex]\[
\sqrt[3]{8} = 2
\][/tex]
- The cube root of 5 remains [tex]\(\sqrt[3]{5}\)[/tex] or [tex]\(5^{1/3}\)[/tex].
- The cube root of [tex]\(w^8\)[/tex]:
[tex]\[
\sqrt[3]{w^8} = w^{8/3}
\][/tex]
4. Combine the simplified parts:
[tex]\[
2 \times 5^{1/3} \times w^{8/3}
\][/tex]
So, the simplified radical form of [tex]\( \sqrt[3]{40 w^8} \)[/tex] is:
[tex]\[
2 \cdot 5^{1/3} \cdot w^{8/3}
\][/tex]
And expressing the exponent [tex]\(8/3\)[/tex] as a decimal, the final result can be written as:
[tex]\[
2 \times 5^{1/3} \times w^{2.66666666666667}
\][/tex]
