Let's solve for the product of the given terms step-by-step.
First, consider the product [tex]\(\sqrt[3]{x^2} \cdot \sqrt[4]{x^3}\)[/tex].
1. Simplifying Each Term:
- The first term is [tex]\(\sqrt[3]{x^2}\)[/tex], which can be expressed as [tex]\(x^{2/3}\)[/tex].
- The second term is [tex]\(\sqrt[4]{x^3}\)[/tex], which can be expressed as [tex]\(x^{3/4}\)[/tex].
2. Product of the Terms:
Next, we compute the product of these terms:
[tex]\[
x^{2/3} \cdot x^{3/4}
\][/tex]
3. Combining Exponents:
When multiplying terms with the same base, we add the exponents:
[tex]\[
x^{2/3} \cdot x^{3/4} = x^{2/3 + 3/4}
\][/tex]
4. Finding a Common Denominator:
To add the exponents [tex]\(2/3\)[/tex] and [tex]\(3/4\)[/tex], find a common denominator. The least common multiple of 3 and 4 is 12. So,
[tex]\[
\frac{2}{3} = \frac{2 \cdot 4}{3 \cdot 4} = \frac{8}{12}
\][/tex]
[tex]\[
\frac{3}{4} = \frac{3 \cdot 3}{4 \cdot 3} = \frac{9}{12}
\][/tex]
5. Adding Exponents:
Now add the two fractions:
[tex]\[
\frac{8}{12} + \frac{9}{12} = \frac{8 + 9}{12} = \frac{17}{12}
\][/tex]
6. Simplified Product:
Thus, the initial product simplifies to:
[tex]\[
x^{17/12}
\][/tex]
Now, let's use the result from above ([tex]\(x^{17/12}\)[/tex]) to evaluate the remaining expressions.
1. Evaluate [tex]\( x \sqrt{x} \)[/tex]:
[tex]\[
x \sqrt{x} = x \cdot x^{1/2} = x^{1 + 1/2} = x^{3/2}
\][/tex]
2. Compute [tex]\(\sqrt[12]{x^5}\)[/tex]:
[tex]\[
\sqrt[12]{x^5} = x^{5/12}
\][/tex]
3. Compute [tex]\(x \left( \sqrt[12]{x^5} \right) \)[/tex]:
[tex]\[
x \left( \sqrt[12]{x^5} \right) = x \cdot x^{5/12} = x^{1 + 5/12} = x^{17/12}
\][/tex]
Finally, the expression [tex]\(x^6\)[/tex] is clearly independent of the other computations, but for completeness, [tex]\(x^6\)[/tex] remains as is.
Based on these calculations, the simplified result for the product matches:
[tex]\[
x \left( \sqrt[12]{x^5} \right)
\][/tex]
Thus, the product [tex]\(\sqrt[3]{x^2} \cdot \sqrt[4]{x^3}\)[/tex] simplifies to:
[tex]\[
x \left( \sqrt[12]{x^5} \right)
\][/tex]
