Equivalent Expression For The Fourth Root Of X To The Tenth Power

At the heart of algebra lies the fascinating interplay between radicals and exponents. Understanding how to manipulate these mathematical concepts is crucial for simplifying expressions and solving equations. In this comprehensive exploration, we will delve into the intricacies of finding the expression equivalent to the fourth root of x to the tenth power, $\sqrt[4]{x^{10}}$. This journey will not only equip you with the necessary tools to tackle similar problems but also illuminate the fundamental principles that govern the world of radicals and exponents.

When faced with an expression like $\sqrt[4]{x^{10}}$, the initial step involves deciphering the meaning of the radical notation. The fourth root, denoted by the index 4, signifies the inverse operation of raising to the fourth power. In essence, we seek a value that, when raised to the fourth power, yields $x^{10}$. To unravel this, we can leverage the fundamental relationship between radicals and fractional exponents. The expression $\sqrt[n]{a^m}$ can be equivalently expressed as $a^{\frac{m}{n}}$. Applying this principle to our problem, we transform $\sqrt[4]{x^{10}}$ into $x^{\frac{10}{4}}$. This transformation marks a pivotal point in our quest, as it allows us to wield the power of exponent rules to simplify the expression.

The fractional exponent $\frac{10}{4}$ can be simplified to $\frac{5}{2}$ by dividing both the numerator and denominator by their greatest common divisor, which is 2. This simplification unveils a clearer picture of the expression's structure: $x^{\frac{5}{2}}$. Now, we can decompose the fractional exponent into its integer and fractional parts. The fraction $\frac{5}{2}$ can be expressed as the sum of 2 and $\frac{1}{2}$. This decomposition allows us to rewrite $x^{\frac{5}{2}}$ as $x^{2 + \frac{1}{2}}$. Here, we invoke another fundamental exponent rule: $a^{m+n} = a^m \cdot a^n$. Applying this rule, we transform $x^{2 + \frac{1}{2}}$ into $x^2 \cdot x^{\frac{1}{2}}$. This step elegantly separates the integer exponent from the fractional exponent, paving the way for further simplification.

Now, we circle back to the relationship between fractional exponents and radicals. Recall that $a^{\frac{1}{n}}$ is equivalent to $\sqrt[n]{a}$. Therefore, $x^{\frac{1}{2}}$ is equivalent to $\sqrt{x}$. Substituting this equivalence into our expression, we arrive at $x^2 \cdot \sqrt{x}$. This expression represents a simplified form of the original expression, $\sqrt[4]{x^{10}}$. However, let's explore another avenue to express the simplified form that aligns with the given answer choices. We initially transformed the fourth root of $x^{10}$ into $x^{\frac{10}{4}}$, which simplified to $x^{\frac{5}{2}}$. An alternative decomposition of the fraction $\frac{5}{2}$ is $2 + \frac{2}{4}$. This might seem like a detour, but it leads to a crucial connection. Rewriting $x^{\frac{5}{2}}$ as $x^{2 + \frac{2}{4}}$ and applying the exponent rule $a^{m+n} = a^m \cdot a^n$, we get $x^2 \cdot x^{\frac{2}{4}}$. Now, we simplify the fractional exponent $\frac{2}{4}$ to $\frac{1}{2}$, giving us $x^2 \cdot x^{\frac{1}{2}}$. Transforming the fractional exponent back into radical form, we have $x^2 \cdot \sqrt[4]{x^2}$. This expression elegantly matches one of the answer choices, solidifying our solution.

Now, let's meticulously examine the answer choices provided and determine which one aligns perfectly with our derived equivalent expression.

A. $x^2(\sqrt[4]{x^2})$

This answer choice resonates strongly with our derived expression. As we meticulously demonstrated in the previous section, $\sqrt[4]{x^{10}}$ can indeed be simplified to $x^2(\sqrt[4]{x^2})$. This equivalence stems from the fundamental interplay between fractional exponents and radicals. By expressing the fourth root as a fractional exponent, simplifying the fraction, and then strategically decomposing the exponent, we arrived at this very form. Therefore, answer choice A stands as a strong contender.

B. $x^{2.2}$

This answer choice presents a decimal exponent, which warrants careful consideration. While it might seem tempting to directly convert $\frac{5}{2}$ to 2.5, it's crucial to recognize that this conversion can obscure the underlying structure of the expression. To rigorously compare this option, we can rewrite 2.5 as a fraction: $\frac{11}{5}$. This allows us to express $x^{2.2}$ as $x^{\frac{11}{5}}$. Converting this back to radical form gives us $\sqrt[5]{x^{11}}$. This expression is distinctly different from our original expression, $\sqrt[4]{x^{10}}$, and our derived equivalent form, $x^2(\sqrt[4]{x^2})$. Therefore, answer choice B does not hold the key to our solution.

C. $x^3(\sqrt[4]{x})$

This answer choice introduces a different power of x outside the radical and a different power of x inside the radical. To assess its validity, we can convert the radical back to a fractional exponent and combine the terms. Expressing $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$, we can rewrite the expression as $x^3 \cdot x^{\frac{1}{4}}$. Invoking the exponent rule $a^m \cdot a^n = a^{m+n}$, we combine the terms to get $x^{3 + \frac{1}{4}}$. Simplifying the exponent, we have $x^{\frac{13}{4}}$. Converting this back to radical form, we obtain $\sqrt[4]{x^{13}}$. This expression deviates from our original expression, $\sqrt[4]{x^{10}}$, and our derived equivalent form, $x^2(\sqrt[4]{x^2})$. Therefore, answer choice C does not align with our solution.

D. $x^5$

This answer choice presents a simple power of x, which might appear enticing at first glance. However, a closer examination reveals that it does not capture the essence of the fourth root. To evaluate its correctness, we can raise $x^5$ to the fourth power and check if it yields $x^{10}$. Raising $x^5$ to the fourth power gives us $(x^5)^4 = x^{20}$, which is significantly different from $x^{10}$. Therefore, answer choice D does not represent an equivalent expression.

Through our meticulous analysis and rigorous examination of the answer choices, we have definitively established that A. $x^2(\sqrt[4]{x^2})$ is the expression equivalent to $\sqrt[4]{x^{10}}$. This equivalence is rooted in the fundamental principles governing radicals and exponents, which we have explored in detail throughout this discussion. By converting the radical to a fractional exponent, simplifying the exponent, and strategically decomposing it, we successfully unveiled the equivalent expression.

To further solidify your understanding of this concept, let's consider a few additional examples:

1. Simplify $\sqrt[3]{x^7}$

   • Convert to fractional exponent: $x^{\frac{7}{3}}$
   • Decompose the exponent: $x^{2 + \frac{1}{3}}$
   • Apply exponent rule: $x^2 \cdot x^{\frac{1}{3}}$
   • Convert back to radical form: $x^2\sqrt[3]{x}$

2. Simplify $\sqrt[5]{x^{12}}$

   • Convert to fractional exponent: $x^{\frac{12}{5}}$
   • Decompose the exponent: $x^{2 + \frac{2}{5}}$
   • Apply exponent rule: $x^2 \cdot x^{\frac{2}{5}}$
   • Convert back to radical form: $x^2\sqrt[5]{x^2}$

3. Simplify $\sqrt[4]{x^9}$

   • Convert to fractional exponent: $x^{\frac{9}{4}}$
   • Decompose the exponent: $x^{2 + \frac{1}{4}}$
   • Apply exponent rule: $x^2 \cdot x^{\frac{1}{4}}$
   • Convert back to radical form: $x^2\sqrt[4]{x}$

By working through these examples, you can reinforce your grasp of the process involved in simplifying radical expressions and identifying equivalent forms.

The journey through simplifying $\sqrt[4]{x^{10}}$ has illuminated the profound connection between radicals and exponents. This understanding extends far beyond this specific problem, empowering you to tackle a wide array of algebraic challenges. By mastering the art of converting between radicals and fractional exponents, simplifying expressions, and strategically applying exponent rules, you unlock a powerful toolkit for navigating the world of mathematics. So, embrace the beauty and versatility of radicals and exponents, and let them guide you towards mathematical mastery.