Simplifying The Expression $\sqrt[6]{x^{65}}$ For X Greater Than 0
#Understanding the Expression $\sqrt[6]{x^{65}}$
In the realm of mathematics, simplifying expressions with radicals and exponents is a fundamental skill. This article delves into simplifying the expression $\sqrt[6]{x^{65}}$, given that $x \textgreater 0$. Our objective is to find an equivalent form among the provided options. To navigate through this problem effectively, we need to understand the properties of exponents and radicals, and how they interact with each other. The expression $\sqrt[6]{x^{65}}$ involves a sixth root and a power of 65, suggesting that we can potentially extract whole powers of x from the radical while leaving a remainder inside. This process is akin to dividing the exponent 65 by the index of the root, which is 6. The quotient will represent the whole power of x that can be taken out, and the remainder will represent the power of x that stays inside the radical. Let's break down the concept further. The expression $\sqrt[n]{x^m}$ can be rewritten using fractional exponents as $x^{\frac{m}{n}}$. In our case, $\sqrt[6]{x^{65}}$ is equivalent to $x^{\frac{65}{6}}$. To simplify this, we can divide 65 by 6. This division yields a quotient of 10 and a remainder of 5. This means that $\frac{65}{6}$ can be expressed as 10 + $\frac{5}{6}$. Now we can rewrite $x^{\frac{65}{6}}$ as $x^{10 + \frac{5}{6}}$. Using the properties of exponents, specifically the rule $x^{a+b} = x^a \cdot x^b$, we can further break this down into $x^{10} \cdot x^{\frac{5}{6}}$. Converting the fractional exponent back to a radical, we get $x^{\frac{5}{6}} = \sqrt[6]{x^5}$. Therefore, the original expression $\sqrt[6]{x^{65}}$ simplifies to $x^{10} \cdot \sqrt[6]{x^5}$. This form allows us to clearly see the whole power of x extracted from the radical and the remaining power of x under the sixth root. This methodical approach is crucial in simplifying such expressions and arriving at the correct equivalent form. By understanding the interplay between exponents and radicals, we can confidently tackle similar problems and deepen our understanding of algebraic manipulations. The initial step in simplifying the expression $\sqrt[6]{x^{65}}$ involves recognizing the relationship between radicals and fractional exponents. Specifically, the expression $\sqrt[6]{x^{65}}$ can be rewritten as $x^{\frac{65}{6}}$. This transformation is crucial because it allows us to apply the rules of exponents more directly. The fraction $\frac{65}{6}$ represents the exponent to which x is raised. To simplify this, we need to break down the fraction into its whole number and fractional parts. Dividing 65 by 6 gives us a quotient of 10 and a remainder of 5. This means that $\frac{65}{6}$ can be expressed as 10 + $\frac{5}{6}$. Now we can rewrite the expression $x^{\frac{65}{6}}$ as $x^{10 + \frac{5}{6}}$. The next step is to apply the rule of exponents that states $x^{a+b} = x^a \cdot x^b$. This rule allows us to separate the exponent into its whole number and fractional parts, resulting in $x^{10} \cdot x^{\frac{5}{6}}$. This separation is key to extracting the whole power of x from the radical. The term $x^{10}$ represents the whole power of x that can be taken out of the sixth root, while the term $x^{\frac{5}{6}}$ represents the remaining power of x that will stay inside the radical. To finalize the simplification, we need to convert the fractional exponent $x^{\frac{5}{6}}$ back into radical form. Recall that $x^{\frac{m}{n}} = \sqrt[n]{x^m}$. Applying this rule, we get $x^{\frac{5}{6}} = \sqrt[6]{x^5}$. Therefore, the original expression $\sqrt[6]{x^{65}}$ simplifies to $x^{10} \cdot \sqrt[6]{x^5}$. This final form clearly shows the simplified expression with the whole power of x and the remaining radical term. In conclusion, the simplification process involves converting the radical to a fractional exponent, breaking the exponent into whole and fractional parts, applying the rules of exponents, and converting the fractional exponent back to radical form. This methodical approach ensures that we arrive at the correct simplified expression. By mastering these techniques, we can confidently handle complex expressions involving radicals and exponents.
#Detailed Solution to Simplifying $\sqrt[6]{x^{65}}$
To tackle the problem of simplifying the expression $\sqrt[6]{x^{65}}$, where $x \textgreater 0$, we will employ a systematic approach that leverages the properties of exponents and radicals. The core idea is to rewrite the radical expression using fractional exponents, simplify the exponent, and then convert back to radical form if necessary. The initial expression we have is $\sqrt[6]{x^{65}}$. The first step is to convert this radical expression into an exponential form. Recall that the nth root of a number raised to the mth power can be written as $x^{\frac{m}{n}}$. Applying this to our expression, we get $\sqrt[6]{x^{65}} = x^{\frac{65}{6}}$. This conversion is crucial because it allows us to manipulate the expression using the rules of exponents. Next, we need to simplify the exponent $ rac{65}{6}$. To do this, we perform division to find the quotient and remainder. Dividing 65 by 6 gives us a quotient of 10 and a remainder of 5. This means that $ rac{65}{6} = 10 + \frac{5}{6}$. Now we can rewrite the expression $x^{\frac{65}{6}}$ as $x^{10 + \frac{5}{6}}$. The next step involves using the property of exponents that states $x^{a+b} = x^a \cdot x^b$. Applying this rule, we get $x^{10 + \frac{5}{6}} = x^{10} \cdot x^{\frac{5}{6}}$. This separation is essential because it isolates the whole number exponent from the fractional exponent. The term $x^{10}$ represents a whole power of x, while the term $x^{\frac{5}{6}}$ represents the remaining fractional power that will be converted back into a radical. Now we need to convert the fractional exponent $x^{\frac{5}{6}}$ back into radical form. Recall that $x^{\frac{m}{n}} = \sqrt[n]{x^m}$. Applying this rule, we get $x^{\frac{5}{6}} = \sqrt[6]{x^5}$. Therefore, the expression $x^{10} \cdot x^{\frac{5}{6}}$ becomes $x^{10} \cdot \sqrt[6]{x^5}$. Combining these steps, we find that the original expression $\sqrt[6]{x^{65}}$ simplifies to $x^{10} \sqrt[6]{x^5}$. This is the final simplified form of the expression. By systematically applying the properties of exponents and radicals, we have successfully simplified the given expression. This approach highlights the importance of understanding the relationship between exponents and radicals and how to manipulate them to simplify complex expressions. In summary, the process involves converting the radical to a fractional exponent, simplifying the exponent by finding the quotient and remainder, applying the rules of exponents to separate the whole and fractional parts, and converting the fractional exponent back to radical form. This step-by-step method ensures accuracy and clarity in simplifying such expressions.
#Identifying the Correct Equivalent Form
After simplifying the expression $\sqrt[6]{x^{65}}$, we arrived at the form $x^{10} \sqrt[6]{x^5}$. Now, we need to match this simplified form with the given options to identify the correct equivalent expression. The options provided are:
A. $x^{10} \sqrt[6]{x^6}$ B. $x^9 \sqrt[6]{x^6}$ C. $x^9 \sqrt[6]{x^5}$ D. $x^{10} \sqrt[6]{x^5}$
Let's compare our simplified expression with each option:
Option A: $x^{10} \sqrt[6]{x^6}$
This option has $x^{10}$ outside the radical, which matches our simplified expression. However, the term inside the radical is $\sqrt[6]{x^6}$. Since $\sqrt[6]{x^6} = x$, this option simplifies to $x^{10} \cdot x = x^{11}$. This does not match our simplified form of $x^{10} \sqrt[6]{x^5}$, so option A is incorrect.
Option B: $x^9 \sqrt[6]{x^6}$
This option has $x^9$ outside the radical, which does not match our simplified expression's $x^{10}$. Similar to option A, the term inside the radical is $\sqrt[6]{x^6}$, which simplifies to x. Thus, this option simplifies to $x^9 \cdot x = x^{10}$. This also does not match our simplified form, so option B is incorrect.
Option C: $x^9 \sqrt[6]{x^5}$
This option has $x^9$ outside the radical, which again does not match our simplified expression's $x^{10}$. The term inside the radical is $\sqrt[6]{x^5}$, which does match the radical part of our simplified expression. However, since the term outside the radical is incorrect, option C is incorrect.
Option D: $x^{10} \sqrt[6]{x^5}$
This option has $x^{10}$ outside the radical, which matches our simplified expression. The term inside the radical is $\sqrt[6]{x^5}$, which also matches the radical part of our simplified expression. Therefore, option D matches our simplified expression exactly.
Based on this comparison, we can conclude that the correct equivalent form of $\sqrt[6]x^{65}}$ is option D \sqrt[6]{x^5}$. This methodical comparison ensures that we have correctly identified the equivalent expression by matching both the term outside the radical and the term inside the radical with our simplified form. By carefully examining each option, we can confidently select the correct answer and reinforce our understanding of simplifying radical expressions.
#Conclusion: The Equivalent Expression
In conclusion, by systematically simplifying the expression $\sqrt[6]{x^{65}}$, we have determined that its equivalent form is $x^{10} \sqrt[6]{x^5}$. This process involved converting the radical expression to an exponential form, simplifying the exponent, separating the whole and fractional parts, and then converting the fractional exponent back to radical form. This detailed exploration underscores the importance of understanding and applying the properties of exponents and radicals in algebraic manipulations. The initial step was to recognize that $\sqrt[6]{x^{65}}$ can be rewritten as $x^{\frac{65}{6}}$. This conversion is fundamental because it allows us to apply the rules of exponents more directly. The fraction $\frac{65}{6}$ was then simplified by dividing 65 by 6, which yielded a quotient of 10 and a remainder of 5. This allowed us to express $\frac{65}{6}$ as 10 + $\frac{5}{6}$. Next, we used the property of exponents $x^{a+b} = x^a \cdot x^b$ to rewrite $x^{\frac{65}{6}}$ as $x^{10} \cdot x^{\frac{5}{6}}$. This separation is crucial because it isolates the whole power of x from the fractional power. The term $x^{10}$ represents the whole power of x that can be taken out of the radical, while the term $x^{\frac{5}{6}}$ represents the remaining power of x that will stay inside the radical. Finally, we converted the fractional exponent $x^{\frac{5}{6}}$ back into radical form, resulting in $\sqrt[6]{x^5}$. Thus, the simplified expression is $x^{10} \sqrt[6]{x^5}$. This methodical approach ensures that we arrive at the correct simplified form. By understanding the relationship between exponents and radicals, we can confidently tackle similar problems and deepen our understanding of algebraic manipulations. The comparison with the given options further solidified our conclusion. Option D, $x^{10} \sqrt[6]{x^5}$, was the only option that matched our simplified expression exactly. The other options either had an incorrect power of x outside the radical or an incorrect term inside the radical. Therefore, the correct equivalent form of $\sqrt[6]{x^{65}}$ is indeed $x^{10} \sqrt[6]{x^5}$. This comprehensive analysis highlights the importance of a systematic approach in simplifying mathematical expressions. By breaking down the problem into smaller, manageable steps, we can effectively navigate complex expressions and arrive at the correct solution.
