Sure, let's solve each part of the problem step-by-step.
Step 1: Calculate [tex]\( 5(\sqrt[3]{x}) + 9(\sqrt[3]{x}) \)[/tex].
When we add these terms together, we combine the coefficients of [tex]\(\sqrt[3]{x}\)[/tex]:
[tex]\[
5(\sqrt[3]{x}) + 9(\sqrt[3]{x}) = (5 + 9)(\sqrt[3]{x}) = 14(\sqrt[3]{x})
\][/tex]
Thus,
[tex]\[
5(\sqrt[3]{x}) + 9(\sqrt[3]{x}) = 14x^{1/3}
\][/tex]
Step 2: Calculate [tex]\( 14(\sqrt[6]{x}) \)[/tex].
This is already in simplified form:
[tex]\[
14(\sqrt[6]{x}) = 14x^{1/6}
\][/tex]
Step 3: Calculate [tex]\( 14\left(\sqrt[6]{x^2}\right) \)[/tex].
We simplify the expression [tex]\(\sqrt[6]{x^2}\)[/tex], which means taking [tex]\(x^2\)[/tex] to the power of [tex]\(\frac{1}{6}\)[/tex]:
[tex]\[
\sqrt[6]{x^2} = (x^2)^{1/6} = x^{2/6} = x^{1/3}
\][/tex]
So,
[tex]\[
14\left(\sqrt[6]{x^2}\right) = 14x^{1/3}
\][/tex]
Step 4: Calculate [tex]\( 14(\sqrt[3]{x}) \)[/tex].
This is already in simplified form:
[tex]\[
14(\sqrt[3]{x}) = 14x^{1/3}
\][/tex]
Step 5: Calculate [tex]\( 14\left(\sqrt[3]{x^2}\right) \)[/tex].
We simplify the expression [tex]\(\sqrt[3]{x^2}\)[/tex], which means taking [tex]\(x^2\)[/tex] to the power of [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\sqrt[3]{x^2} = (x^2)^{1/3} = x^{2/3}
\][/tex]
So,
[tex]\[
14\left(\sqrt[3]{x^2}\right) = 14x^{2/3}
\][/tex]
To summarize, the calculated expressions are:
1. [tex]\( 5(\sqrt[3]{x}) + 9(\sqrt[3]{x}) = 14x^{1/3} \)[/tex]
2. [tex]\( 14(\sqrt[6]{x}) = 14x^{1/6} \)[/tex]
3. [tex]\( 14\left(\sqrt[6]{x^2}\right) = 14x^{1/3} \)[/tex]
4. [tex]\( 14(\sqrt[3]{x}) = 14x^{1/3} \)[/tex]
5. [tex]\( 14\left(\sqrt[3]{x^2}\right) = 14x^{2/3} \)[/tex]
Therefore, the resulting terms are:
[tex]\[
(14x^{1/3}, 14x^{1/6}, 14x^{1/3}, 14x^{1/3}, 14x^{2/3})
\][/tex]
