Answer :
To find the product of the given expressions, we need to carefully handle the exponents and logarithmic properties. Let’s break this down step-by-step:
### Given Expressions:
1. [tex]\(\sqrt[3]{x^2}\)[/tex]
2. [tex]\(\sqrt[4]{x^3}\)[/tex]
3. [tex]\(x \sqrt{x}\)[/tex]
4. [tex]\(\sqrt[12]{x^5}\)[/tex]
5. [tex]\(x\left(\sqrt[12]{x^5}\right)\)[/tex]
6. [tex]\(x^6\)[/tex]
### Converting Root Expressions to Exponents:
We can convert each root expression to its corresponding exponent form:
1. [tex]\(\sqrt[3]{x^2}\)[/tex] can be rewritten as [tex]\(x^{\frac{2}{3}}\)[/tex]
2. [tex]\(\sqrt[4]{x^3}\)[/tex] can be rewritten as [tex]\(x^{\frac{3}{4}}\)[/tex]
3. [tex]\(x \sqrt{x}\)[/tex] can be rewritten as [tex]\(x \cdot x^{\frac{1}{2}} = x^{1 + \frac{1}{2}} = x^{\frac{3}{2}}\)[/tex]
4. [tex]\(\sqrt[12]{x^5}\)[/tex] can be rewritten as [tex]\(x^{\frac{5}{12}}\)[/tex]
5. [tex]\(x\left(\sqrt[12]{x^5}\right)\)[/tex] can be rewritten as [tex]\(x \cdot x^{\frac{5}{12}} = x^{1 + \frac{5}{12}} = x^{\frac{17}{12}}\)[/tex]
6. [tex]\(x^6\)[/tex] remains as [tex]\(x^6\)[/tex]
### Combining the Exponents:
We now multiply all these exponentiated terms together:
[tex]\[ x^{\frac{2}{3}} \cdot x^{\frac{3}{4}} \cdot x^{\frac{3}{2}} \cdot x^{\frac{5}{12}} \cdot x^{\frac{17}{12}} \cdot x^6 \][/tex]
To combine the exponents, we use the rule for multiplying exponents with the same base: [tex]\(x^a \cdot x^b = x^{a+b}\)[/tex]. Therefore, we add all the exponents together:
[tex]\[ \frac{2}{3} + \frac{3}{4} + \frac{3}{2} + \frac{5}{12} + \frac{17}{12} + 6 \][/tex]
First, let's convert all the fractions to have a common denominator, which in this case is 12:
- [tex]\(\frac{2}{3} = \frac{8}{12}\)[/tex]
- [tex]\(\frac{3}{4} = \frac{9}{12}\)[/tex]
- [tex]\(\frac{3}{2} = \frac{18}{12}\)[/tex]
- [tex]\(\frac{5}{12} = \frac{5}{12}\)[/tex]
- [tex]\(\frac{17}{12} = \frac{17}{12}\)[/tex]
- [tex]\(6 = \frac{72}{12}\)[/tex]
Adding these fractions together:
[tex]\[ \frac{8}{12} + \frac{9}{12} + \frac{18}{12} + \frac{5}{12} + \frac{17}{12} + \frac{72}{12} = \frac{129}{12} \][/tex]
This simplifies to [tex]\(x^{\frac{129}{12}}\)[/tex]. Converting [tex]\(\frac{129}{12}\)[/tex] to a mixed fraction gives:
[tex]\[ \frac{129}{12} = 10 \frac{9}{12} = 10.75 \][/tex]
So, the final exponent is [tex]\(x^{10.75}\)[/tex].
### Final Result in Different Forms:
The product is:
1. [tex]\(x^{\frac{129}{12}} \approx x^{10.75}\)[/tex]
2. In root form with fractional exponent: [tex]\( x^{1.4166666666666667}\)[/tex]
3. Representing the root aspect separately, we would also have:
[tex]\[ x \cdot \sqrt{x} \cdot \sqrt[12]{x^5} = x x^{0.5} x^{5/12} = x^{1+0.5+5/12} = x^{17/12} = x^{1.41666666666666667} \][/tex]
Thus, the product of all the given expressions results in the final form:
[tex]\[ x^{\frac{129}{12}}, \quad \text{or} \quad x^{10.75}, \quad \text{or} \quad x \cdot \sqrt{x} \cdot \sqrt[12]{x^5} \text{ and other equivalent forms.} \][/tex]
The complete product of all terms:
[tex]\( x^{\frac{129}{12}}\approx x^{10.75}.\)[/tex]
### Given Expressions:
1. [tex]\(\sqrt[3]{x^2}\)[/tex]
2. [tex]\(\sqrt[4]{x^3}\)[/tex]
3. [tex]\(x \sqrt{x}\)[/tex]
4. [tex]\(\sqrt[12]{x^5}\)[/tex]
5. [tex]\(x\left(\sqrt[12]{x^5}\right)\)[/tex]
6. [tex]\(x^6\)[/tex]
### Converting Root Expressions to Exponents:
We can convert each root expression to its corresponding exponent form:
1. [tex]\(\sqrt[3]{x^2}\)[/tex] can be rewritten as [tex]\(x^{\frac{2}{3}}\)[/tex]
2. [tex]\(\sqrt[4]{x^3}\)[/tex] can be rewritten as [tex]\(x^{\frac{3}{4}}\)[/tex]
3. [tex]\(x \sqrt{x}\)[/tex] can be rewritten as [tex]\(x \cdot x^{\frac{1}{2}} = x^{1 + \frac{1}{2}} = x^{\frac{3}{2}}\)[/tex]
4. [tex]\(\sqrt[12]{x^5}\)[/tex] can be rewritten as [tex]\(x^{\frac{5}{12}}\)[/tex]
5. [tex]\(x\left(\sqrt[12]{x^5}\right)\)[/tex] can be rewritten as [tex]\(x \cdot x^{\frac{5}{12}} = x^{1 + \frac{5}{12}} = x^{\frac{17}{12}}\)[/tex]
6. [tex]\(x^6\)[/tex] remains as [tex]\(x^6\)[/tex]
### Combining the Exponents:
We now multiply all these exponentiated terms together:
[tex]\[ x^{\frac{2}{3}} \cdot x^{\frac{3}{4}} \cdot x^{\frac{3}{2}} \cdot x^{\frac{5}{12}} \cdot x^{\frac{17}{12}} \cdot x^6 \][/tex]
To combine the exponents, we use the rule for multiplying exponents with the same base: [tex]\(x^a \cdot x^b = x^{a+b}\)[/tex]. Therefore, we add all the exponents together:
[tex]\[ \frac{2}{3} + \frac{3}{4} + \frac{3}{2} + \frac{5}{12} + \frac{17}{12} + 6 \][/tex]
First, let's convert all the fractions to have a common denominator, which in this case is 12:
- [tex]\(\frac{2}{3} = \frac{8}{12}\)[/tex]
- [tex]\(\frac{3}{4} = \frac{9}{12}\)[/tex]
- [tex]\(\frac{3}{2} = \frac{18}{12}\)[/tex]
- [tex]\(\frac{5}{12} = \frac{5}{12}\)[/tex]
- [tex]\(\frac{17}{12} = \frac{17}{12}\)[/tex]
- [tex]\(6 = \frac{72}{12}\)[/tex]
Adding these fractions together:
[tex]\[ \frac{8}{12} + \frac{9}{12} + \frac{18}{12} + \frac{5}{12} + \frac{17}{12} + \frac{72}{12} = \frac{129}{12} \][/tex]
This simplifies to [tex]\(x^{\frac{129}{12}}\)[/tex]. Converting [tex]\(\frac{129}{12}\)[/tex] to a mixed fraction gives:
[tex]\[ \frac{129}{12} = 10 \frac{9}{12} = 10.75 \][/tex]
So, the final exponent is [tex]\(x^{10.75}\)[/tex].
### Final Result in Different Forms:
The product is:
1. [tex]\(x^{\frac{129}{12}} \approx x^{10.75}\)[/tex]
2. In root form with fractional exponent: [tex]\( x^{1.4166666666666667}\)[/tex]
3. Representing the root aspect separately, we would also have:
[tex]\[ x \cdot \sqrt{x} \cdot \sqrt[12]{x^5} = x x^{0.5} x^{5/12} = x^{1+0.5+5/12} = x^{17/12} = x^{1.41666666666666667} \][/tex]
Thus, the product of all the given expressions results in the final form:
[tex]\[ x^{\frac{129}{12}}, \quad \text{or} \quad x^{10.75}, \quad \text{or} \quad x \cdot \sqrt{x} \cdot \sqrt[12]{x^5} \text{ and other equivalent forms.} \][/tex]
The complete product of all terms:
[tex]\( x^{\frac{129}{12}}\approx x^{10.75}.\)[/tex]
