Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
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]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
