Simplifying Radicals Expressing \(\frac{\sqrt[3]{4}}{\sqrt{2}}\) In Simplest Radical Form

In the realm of mathematics, simplifying expressions to their most basic form is a crucial skill. This article delves into the process of expressing the fraction ${\frac{\sqrt[3]{4}}{\sqrt{2}}}$ in its simplest radical form. We will explore the underlying concepts of radicals, exponents, and fractions, and utilize them to transform the given expression into its most concise representation. Let's embark on this mathematical journey together!

Understanding Radicals and Exponents

Before we dive into simplifying the given expression, it's essential to have a firm grasp of radicals and exponents. Radicals, often denoted by the symbol ${\sqrt[n]{a}}$, represent the ${n}$-th root of a number ${a}$. The index ${n}$ indicates the degree of the root, while the radicand ${a}$ is the number under the radical sign. For instance, ${\sqrt{9}}$ represents the square root of 9, which is 3, and ${\sqrt[3]{8}}$ signifies the cube root of 8, which is 2.

Exponents, on the other hand, provide a concise way to express repeated multiplication. The expression ${a^n}$ indicates that the base ${a}$ is multiplied by itself ${n}$ times. For example, ${2^3}$ is equivalent to 2 × 2 × 2, which equals 8. Exponents also play a crucial role in representing radicals. The ${n}$-th root of a number ${a}$ can be expressed as ${a^{\frac{1}{n}}}$. This connection between radicals and exponents is fundamental to simplifying radical expressions.

Converting Radicals to Exponential Form

The ability to convert between radical and exponential forms is a powerful tool in simplifying radical expressions. Let's consider the given expression, ${\frac{\sqrt[3]{4}}{\sqrt{2}}}$ . We can rewrite the cube root of 4 as ${4^{\frac{1}{3}}}$ and the square root of 2 as ${2^{\frac{1}{2}}}$ . This transformation allows us to work with exponents, which are often easier to manipulate than radicals directly. The expression now becomes ${\frac{4^{\frac{1}{3}}}{2^{\frac{1}{2}}}}$ . This conversion is a crucial first step in our simplification process, as it sets the stage for applying exponent rules and algebraic manipulations.

Simplifying the Expression

With the expression now in exponential form, we can proceed with simplification. The key to simplifying ${\frac{4^{\frac{1}{3}}}{2^{\frac{1}{2}}}}$ lies in expressing both the numerator and denominator with the same base. Notice that 4 can be written as ${2^2}$. Substituting this into the expression, we get ${\frac{(2^2)^{\frac{1}{3}}}{2^{\frac{1}{2}}}}$ . Now, we can apply the power of a power rule, which states that ${(a^m)^n = a^{mn}}$. Applying this rule to the numerator, we have ${2^{2 \times \frac{1}{3}} = 2^{\frac{2}{3}}}$ . Our expression now looks like this: ${\frac{2^{\frac{2}{3}}}{2^{\frac{1}{2}}}}$ . This step of expressing both numerator and denominator with the same base is critical, as it allows us to use the quotient rule of exponents.

Applying Exponent Rules

Now that we have a common base, we can utilize the quotient rule of exponents, which states that ${\frac{a^m}{a^n} = a^{m-n}}$. Applying this rule to our expression, ${\frac{2^{\frac{2}{3}}}{2^{\frac{1}{2}}}}$ , we subtract the exponents: ${2^{\frac{2}{3} - \frac{1}{2}}}$ . To subtract the fractions in the exponent, we need a common denominator. The least common multiple of 3 and 2 is 6, so we rewrite the fractions with a denominator of 6: ${\frac{2}{3} = \frac{4}{6}}$ and ${\frac{1}{2} = \frac{3}{6}}$. The exponent now becomes ${\frac{4}{6} - \frac{3}{6} = \frac{1}{6}}$. Therefore, our simplified expression in exponential form is ${2^{\frac{1}{6}}}$ . This step-by-step application of exponent rules is essential to accurately simplify the expression.

Converting Back to Radical Form

While ${2^{\frac{1}{6}}}$ is a simplified form, the question asks for the simplest radical form. To convert back to radical form, we recall that ${a^{\frac{1}{n}}}$ is equivalent to ${\sqrt[n]{a}}$. In our case, ${2^{\frac{1}{6}}}$ translates to ${\sqrt[6]{2}}$. This is the simplest radical form of the given expression. We have successfully transformed the original expression, ${\frac{\sqrt[3]{4}}{\sqrt{2}}}$ , into its most concise radical representation.

Verifying the Simplest Form

It's important to ensure that the radical expression is indeed in its simplest form. A radical is considered simplest when the radicand has no perfect square factors (for square roots), perfect cube factors (for cube roots), and so on, and when the index is as small as possible. In the case of ${\sqrt[6]{2}}$, the radicand, 2, has no factors other than 1 and itself, so it has no perfect sixth root factors. Additionally, the index, 6, cannot be reduced further while maintaining an integer exponent. Therefore, ${\sqrt[6]{2}}$ is indeed the simplest radical form.

Conclusion

In this article, we have successfully simplified the expression ${\frac{\sqrt[3]{4}}{\sqrt{2}}}$ to its simplest radical form, which is ${\sqrt[6]{2}}$. We accomplished this by converting radicals to exponential form, applying exponent rules, and then converting back to radical form. This process highlights the interconnectedness of radicals and exponents and demonstrates how a solid understanding of these concepts is crucial for simplifying mathematical expressions. The ability to manipulate and simplify expressions like this is a fundamental skill in algebra and beyond, paving the way for more advanced mathematical concepts. Remember, the key to simplification lies in breaking down the problem into manageable steps and applying the appropriate rules and techniques.