Simplifying Nested Radicals A Step-by-Step Solution For √(∛((2/5)⁴²))

Navigating the world of mathematical expressions often involves encountering nested radicals, which can initially appear daunting. However, by employing the fundamental principles of exponents and radicals, we can systematically simplify these expressions and reveal their underlying simplicity. In this article, we embark on a journey to unravel the intricacies of the nested radical expression $\sqrt{\sqrt[3]{\left(\frac{2}{5}\right)^{42}}}$, providing a comprehensive, step-by-step solution alongside explanations that cater to both novice and seasoned mathematical enthusiasts.

Understanding the Fundamentals: Exponents and Radicals

Before diving into the specifics of the expression, it's crucial to solidify our understanding of exponents and radicals. Exponents, in their essence, represent repeated multiplication. For instance, $x^n$ signifies multiplying $x$ by itself $n$ times. Conversely, radicals are the inverse operation of exponentiation. The expression $\sqrt[n]{x}$ seeks the number that, when raised to the power of $n$, yields $x$. In the case of a square root ($\sqrt{x}$), we are looking for a number that, when squared, equals $x$. Similarly, a cube root ($\sqrt[3]{x}$) seeks a number that, when cubed, equals $x$.

The relationship between exponents and radicals is elegantly captured by the following equivalence: $\sqrt[n]{x} = x^{\frac{1}{n}}$. This crucial identity allows us to seamlessly transition between radical and exponential forms, often simplifying complex expressions. For example, $\sqrt{x}$ can be rewritten as $x^{\frac{1}{2}}$, and $\sqrt[3]{x}$ becomes $x^{\frac{1}{3}}$. Mastering this conversion is key to effectively manipulating radical expressions.

Furthermore, when dealing with nested radicals, the power of exponents truly shines. Consider an expression like $\sqrt[m]{\sqrt[n]{x}}$. This can be elegantly simplified by applying the exponential conversion twice. First, we rewrite the inner radical as $x^{\frac{1}{n}}$, giving us $\sqrt[m]{x^{\frac{1}{n}}}$. Next, we convert the outer radical, resulting in $\left(x^{\frac{1}{n}}\right)^{\frac{1}{m}}$. A fundamental property of exponents states that $(x^a)^b = x^{a \cdot b}$. Applying this property, we arrive at the simplified form $x^{\frac{1}{n} \cdot \frac{1}{m}} = x^{\frac{1}{nm}}$. This elegantly demonstrates that nested radicals can be simplified by multiplying the indices of the radicals.

With these fundamental concepts firmly in place, we are now well-equipped to tackle the nested radical expression at hand. The key takeaway is the ability to convert between radical and exponential forms and to understand how exponents interact with each other, especially when dealing with nested structures. This foundational knowledge will not only aid in solving this particular problem but also in navigating a wide range of mathematical challenges involving radicals and exponents.

Deconstructing the Expression: $\sqrt{\sqrt[3]{\left(\frac{2}{5}\right)^{42}}}$

Let's now turn our attention to the specific expression: $\sqrt{\sqrt[3]{\left(\frac{2}{5}\right)^{42}}}$. To effectively simplify this nested radical, we will employ a step-by-step approach, leveraging the principles of exponents and radicals discussed earlier. Our primary goal is to eliminate the radicals by converting them into exponential forms and then simplifying the resulting exponents.

Our initial step involves addressing the innermost radical, the cube root. Recall that $\sqrt[n]{x}$ can be expressed as $x^{\frac{1}{n}}$. Applying this to our expression, we rewrite $\sqrt[3]{\left(\frac{2}{5}\right)^{42}}$ as $\left(\frac{2}{5}\right)^{\frac{42}{3}}$. This conversion is crucial as it transforms the radical into a more manageable exponential form. The exponent $\frac{42}{3}$ can be readily simplified to 14, making the expression $\left(\frac{2}{5}\right)^{14}$.

Now, our expression takes the form $\sqrt{\left(\frac{2}{5}\right)^{14}}$. We have successfully eliminated the inner cube root, and we are left with a single square root. To tackle this, we once again apply the principle of converting radicals to exponents. The square root, $\sqrt{x}$, is equivalent to raising $x$ to the power of $\frac{1}{2}$. Therefore, we rewrite $\sqrt{\left(\frac{2}{5}\right)^{14}}$ as $\left(\left(\frac{2}{5}\right)^{14}\right)^{\frac{1}{2}}$.

At this juncture, we encounter another key property of exponents: $(x^a)^b = x^{a \cdot b}$. This rule allows us to simplify the nested exponents by multiplying them. In our case, we have $\left(\left(\frac{2}{5}\right)^{14}\right)^{\frac{1}{2}}$, which simplifies to $\left(\frac{2}{5}\right)^{14 \cdot \frac{1}{2}}$. The product of 14 and $\frac{1}{2}$ is 7, so the expression further simplifies to $\left(\frac{2}{5}\right)^{7}$.

We have successfully navigated the nested radicals and arrived at the simplified form $\left(\frac{2}{5}\right)^{7}$. This result represents the expression without any radicals, making it far easier to understand and interpret. We can also choose to expand this expression by multiplying $\frac{2}{5}$ by itself seven times, but the exponential form is often preferred for its conciseness and clarity.

In summary, by systematically converting radicals to exponents and applying the rules of exponent manipulation, we have elegantly simplified the nested radical expression $\sqrt{\sqrt[3]{\left(\frac{2}{5}\right)^{42}}}$. This process highlights the power of these fundamental mathematical principles in unraveling complex expressions.

The Final Simplification: Calculating $\left(\frac{2}{5}\right)^{7}$

Having simplified the nested radical expression to $\left(\frac{2}{5}\right)^{7}$, we now proceed to calculate the final numerical value. While the exponential form provides a concise representation, expanding it can offer a more concrete understanding of the result. Our task here is to raise the fraction $\frac{2}{5}$ to the power of 7, which means multiplying it by itself seven times.

Recall that when raising a fraction to a power, we raise both the numerator and the denominator to that power. In other words, $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$. Applying this rule to our expression, we get $\left(\frac{2}{5}\right)^{7} = \frac{2^7}{5^7}$. This transformation separates the calculation into two simpler exponentiations.

First, let's calculate $2^7$. This means multiplying 2 by itself seven times: $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$. This can be done step by step: $2^2 = 4$, $2^3 = 8$, $2^4 = 16$, $2^5 = 32$, $2^6 = 64$, and finally, $2^7 = 128$. So, the numerator of our fraction is 128.

Next, we need to calculate $5^7$. This means multiplying 5 by itself seven times: $5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$. This calculation is a bit more involved. We can proceed similarly: $5^2 = 25$, $5^3 = 125$, $5^4 = 625$, $5^5 = 3125$, $5^6 = 15625$, and finally, $5^7 = 78125$. Thus, the denominator of our fraction is 78125.

Combining these results, we have $\left(\frac{2}{5}\right)^{7} = \frac{128}{78125}$. This fraction represents the fully simplified numerical value of the original nested radical expression. While it may appear as a somewhat unwieldy fraction, it is the precise result of our simplification process.

It's worth noting that depending on the context, this fraction could be further converted to a decimal representation. However, the fractional form often provides a more accurate representation, avoiding the potential for rounding errors that can occur with decimals. In this case, the fraction $\frac{128}{78125}$ is already in its simplest form, as 128 and 78125 share no common factors other than 1.

In conclusion, by calculating $2^7$ and $5^7$, we have successfully evaluated $\left(\frac{2}{5}\right)^{7}$ as $\frac{128}{78125}$. This final step completes the simplification of the nested radical expression, providing a clear and precise numerical answer.

Summarizing the Journey: From Nested Radicals to Simplified Form

In this comprehensive exploration, we embarked on a journey to simplify the nested radical expression $\sqrt{\sqrt[3]{\left(\frac{2}{5}\right)^{42}}}$. Our approach was methodical and relied on the fundamental principles of exponents and radicals. We began by establishing a solid foundation in these concepts, emphasizing the crucial relationship between radicals and exponents and how to convert between the two forms.

The first key step in simplifying the expression involved recognizing the equivalence between radicals and fractional exponents. We transformed the cube root into an exponent of $\frac{1}{3}$, allowing us to rewrite the inner radical as $\left(\frac{2}{5}\right)^{\frac{42}{3}}$. Simplifying the exponent, we obtained $\left(\frac{2}{5}\right)^{14}$. This initial conversion was pivotal in unraveling the nested structure.

Next, we tackled the outer square root. Again, we applied the principle of converting radicals to exponents, transforming the square root into an exponent of $\frac{1}{2}$. This gave us $\left(\left(\frac{2}{5}\right)^{14}\right)^{\frac{1}{2}}$. At this stage, we invoked the power of exponent rules, specifically the rule that states $(x^a)^b = x^{a \cdot b}$. By multiplying the exponents, we simplified the expression to $\left(\frac{2}{5}\right)^{7}$.

Having successfully eliminated the radicals, we arrived at a simplified exponential form. To obtain a more concrete understanding of the result, we proceeded to calculate $\left(\frac{2}{5}\right)^{7}$. This involved raising both the numerator and the denominator to the power of 7. We calculated $2^7$ as 128 and $5^7$ as 78125, resulting in the fraction $\frac{128}{78125}$.

Throughout this process, we have highlighted the importance of understanding and applying the fundamental rules of exponents and radicals. The ability to convert between radical and exponential forms is a powerful tool in simplifying complex expressions. Moreover, recognizing and utilizing exponent rules, such as $(x^a)^b = x^{a \cdot b}$, is crucial for efficiently manipulating expressions.

The final result, $\frac{128}{78125}$, represents the fully simplified form of the original nested radical expression. This journey from nested radicals to a simple fraction exemplifies the elegance and power of mathematical simplification. By breaking down the problem into manageable steps and applying the appropriate principles, we have successfully navigated a seemingly complex expression and arrived at a clear and concise solution.

This exploration serves as a testament to the beauty of mathematical problem-solving. By understanding the underlying principles and employing a systematic approach, we can unravel even the most intricate expressions and reveal their inherent simplicity. The skills and techniques demonstrated here are not only applicable to this specific problem but also to a wide range of mathematical challenges involving exponents, radicals, and simplification techniques.