Problem 75 - Intermediate Algebra

Question

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[6]{4} $$

Step-by-Step Solution

Answer

\( 4^{1/6} \)

Step 1: Recognize the Radical Form

The given expression is \( \sqrt[6]{4} \). This can be understood as the 6th root of 4, where 6 is the index of the radical, and 4 is the radicand.

Step 2: Convert to Rational Exponents

A radical \( \sqrt[n]{a} \) can be expressed with rational exponents as \( a^{1/n} \). Therefore, \( \sqrt[6]{4} \) becomes \( 4^{1/6} \).

Step 3: Simplify the Expression

In this specific case, \( 4^{1/6} \) is already in its simplest rational exponent form, as there are no further simplifications possible without additional context or requirements.

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. In a radical expression, you have two parts: the radicand and the index. The radicand is the number or expression inside the radical sign, while the index indicates the degree of the root to extract. For example, in \( \sqrt[6]{4} \), the number 4 is the radicand, and 6 is the index.
Radicals are special because they allow us to express roots of numbers in a simplified form. They are used in numerous mathematical applications, from solving equations to simplifying expressions. Understanding how to express radicals in different formats, such as using rational exponents, is essential for working efficiently in algebra and calculus.
By converting a radical into a rational exponent, like turning \( \sqrt[6]{4} \) into \( 4^{1/6} \), you gain the ability to use exponential rules to manipulate and solve equations more easily.

Simplifying Radicals

Simplifying radicals means reducing them into a simpler or more conventional form, usually by finding equivalent expressions without radicals when possible. Consider the radical \( \sqrt{18} \). It simplifies to \( 3\sqrt{2} \) because 18 is the product of 9 and 2, and 9 is a perfect square.
Steps to simplify radicals:

• Identify perfect squares (or cubes, etc.) in the radicand.
• Express the radicand as a product of its prime factors.
• Rewrite using the perfect squares and simplify.

However, when a radical is already expressed in its simplest form, as with \( 4^{1/6} \), there's often no further simplification possible without more context. Recognizing when a radical is already simplified is a key part of working with these expressions.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \). They include numbers like 1/2, 3, or -4/5. Understanding rational numbers is crucial because they appear in many aspects of math, including in rational exponent expressions.
Rational exponents, as in \( 4^{1/6} \), use roots expressed as exponents. They provide a convenient way to deal with roots and radicals in equations. Just as a rational number simplifies calculations by avoiding awkward decimals, rational exponents simplify complex radical manipulations.By learning how rational numbers connect with radicals through rational exponents, you gain a deeper insight into both fundamental and advanced mathematics. This connection enlarges your toolkit for both simplifying expressions and solving problems effectively.