Simplify $\sqrt[8]{\sqrt[7]{x}}=\sqrt[A]{x}$ Find A A Comprehensive Solution

Introduction

In this article, we will delve into simplifying expressions involving nested radicals. Our primary focus will be on solving the equation $\sqrt[8]{\sqrt[7]{x}}=\sqrt[A]{x}$, where our goal is to find the value of A. This problem falls under the realm of mathematics, specifically dealing with algebraic manipulations and the properties of exponents and radicals. Understanding how to simplify such expressions is crucial for various mathematical applications, making it an essential skill for students and enthusiasts alike.

Understanding Radicals and Exponents

Before we dive into solving the equation, let's revisit the fundamental concepts of radicals and exponents. A radical, denoted by the symbol $\sqrt[n]{x}$, represents the n-th root of x. The number n is called the index of the radical, and x is the radicand. For instance, $\sqrt[2]{9}$ (commonly written as $\sqrt{9}$) represents the square root of 9, which is 3, because 3 * 3 = 9. Similarly, $\sqrt[3]{8}$ represents the cube root of 8, which is 2, because 2 * 2 * 2 = 8. The concept of radicals is inextricably linked to exponents, providing an alternative way to express roots. The n-th root of x can be equivalently written as $x^{\frac{1}{n}}$. This exponential form is immensely useful in simplifying expressions and performing algebraic manipulations.

For example, $\sqrt{x}$ can be written as $x^{\frac{1}{2}}$, and $\sqrt[3]{x}$ can be written as $x^{\frac{1}{3}}$. These notations are interchangeable and can be used based on the context and convenience of the problem at hand. Understanding this equivalence is the first step in simplifying complex expressions involving radicals. When dealing with nested radicals, it becomes even more crucial to leverage the exponential notation, as it allows us to apply the rules of exponents more easily. In our problem, $\sqrt[8]{\sqrt[7]{x}}$, we have a radical nested within another radical, which can be simplified using the exponential form and the properties of exponents.

Converting Radicals to Exponential Form

The cornerstone of simplifying radical expressions lies in converting them into their equivalent exponential forms. This conversion allows us to leverage the well-established rules of exponents, making complex manipulations significantly easier. The general rule for converting a radical to an exponential form is: $\sqrt[n]{x} = x^{\frac{1}{n}}$. This rule states that the n-th root of x is the same as raising x to the power of 1/n. Applying this rule to nested radicals involves a step-by-step approach, working from the innermost radical outwards.

Consider our given expression, $\sqrt[8]{\sqrt[7]{x}}$. The innermost radical is $\sqrt[7]{x}$, which, according to our conversion rule, can be rewritten as $x^{\frac{1}{7}}$. Now, the expression becomes $\sqrt[8]{x^{\frac{1}{7}}}$. We still have a radical to deal with, so we apply the conversion rule again. The outer radical, $\sqrt[8]{}$, can be expressed as raising to the power of $\frac{1}{8}$. Thus, $\sqrt[8]{x^{\frac{1}{7}}}$ becomes $(x^{\frac{1}{7}})^{\frac{1}{8}}$. By converting the radicals to exponential forms, we've transformed the problem into an exercise in applying the rules of exponents, which are much more straightforward to handle. The next step involves using the power of a power rule, which states that $(a^m)^n = a^{m \cdot n}$. This rule is crucial for simplifying expressions with exponents raised to other exponents, as is the case in our transformed expression. Converting to exponential form not only simplifies the notation but also opens the door to using powerful algebraic tools that would be cumbersome to apply directly to radical expressions. This is a fundamental technique in algebra and is essential for solving a wide range of problems.

Applying the Power of a Power Rule

After converting the nested radicals into exponential form, we arrive at the expression $(x^{\frac{1}{7}})^{\frac{1}{8}}$. The key to simplifying this further lies in the application of the power of a power rule. This rule, a fundamental concept in exponents, states that when you raise a power to another power, you multiply the exponents. Mathematically, it is expressed as $(a^m)^n = a^{m \cdot n}$. This rule is particularly useful in situations involving nested exponents or, as in our case, nested radicals converted to exponential form.

Applying the power of a power rule to our expression, we multiply the exponents $\frac{1}{7}$ and $\frac{1}{8}$. This yields: $(x^{\frac{1}{7}})^{\frac{1}{8}} = x^{\frac{1}{7} \cdot \frac{1}{8}}$. To multiply these fractions, we multiply the numerators together and the denominators together: $\frac{1}{7} \cdot \frac{1}{8} = \frac{1 \cdot 1}{7 \cdot 8} = \frac{1}{56}$. Therefore, the expression simplifies to $x^{\frac{1}{56}}$. This transformation is a crucial step in solving our original equation. We've successfully reduced the complex nested radical expression into a single exponent form. The power of a power rule allows us to condense multiple layers of exponents into a single exponent, making the expression much simpler to understand and manipulate. This technique is not only applicable to mathematical problems but also finds its use in various scientific and engineering calculations where exponential relationships are prevalent. Understanding and applying this rule effectively is a cornerstone of algebraic manipulation.

Converting Back to Radical Form and Solving for A

Now that we have simplified the expression $(x^{\frac{1}{7}})^{\frac{1}{8}}$ to $x^{\frac{1}{56}}$, the next step is to convert this exponential form back into radical form. This conversion will allow us to directly compare our simplified expression with the given equation and solve for the unknown variable A. Recall the fundamental relationship between exponents and radicals: $x^{\frac{1}{n}} = \sqrt[n]{x}$. Applying this rule in reverse, we can convert $x^{\frac{1}{56}}$ back into a radical expression. The exponent $\frac{1}{56}$ indicates that we are taking the 56th root of x. Therefore, $x^{\frac{1}{56}}$ is equivalent to $\sqrt[56]{x}$.

Now we can revisit the original equation: $\sqrt[8]{\sqrt[7]{x}}=\sqrt[A]{x}$. We have simplified the left-hand side of the equation to $\sqrt[56]{x}$. Thus, our equation now reads $\sqrt[56]{x} = \sqrt[A]{x}$. For this equation to hold true, the indices of the radicals on both sides must be equal. This means that A must be equal to 56. Therefore, by carefully converting between radical and exponential forms, applying the power of a power rule, and comparing the simplified expression with the original equation, we have successfully determined the value of A. This process highlights the importance of understanding the interplay between different mathematical notations and the power of algebraic manipulation in solving equations.

Conclusion

In conclusion, we have successfully simplified the expression $\sqrt[8]{\sqrt[7]{x}}$ and found the value of A in the equation $\sqrt[8]{\sqrt[7]{x}}=\sqrt[A]{x}$. The process involved converting radicals to exponential form, applying the power of a power rule, and converting back to radical form to solve for A. The key steps were:

1. Converting $\sqrt[7]{x}$ to $x^{\frac{1}{7}}$.
2. Rewriting $\sqrt[8]{\sqrt[7]{x}}$ as $\sqrt[8]{x^{\frac{1}{7}}}$.
3. Converting $\sqrt[8]{x^{\frac{1}{7}}}$ to $(x^{\frac{1}{7}})^{\frac{1}{8}}$.
4. Applying the power of a power rule: $(x^{\frac{1}{7}})^{\frac{1}{8}} = x^{\frac{1}{7} \cdot \frac{1}{8}} = x^{\frac{1}{56}}$.
5. Converting $x^{\frac{1}{56}}$ back to radical form: $\sqrt[56]{x}$.
6. Comparing $\sqrt[56]{x}$ with $\sqrt[A]{x}$ to conclude that A = 56.

This exercise demonstrates the importance of understanding and applying the properties of exponents and radicals in simplifying mathematical expressions. By mastering these techniques, one can tackle more complex problems in algebra and beyond. The ability to convert between different mathematical notations and apply the appropriate rules is a valuable skill in mathematics, and this problem provides a clear example of its application. The solution not only answers the specific question but also reinforces fundamental mathematical principles that are essential for further study in the field.