Problem 63 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( 4^{\frac{1}{6}} \).

Step 1: Understanding the Radicals

The given expression is a sixth root: \( \sqrt[6]{4} \). To simplify this expression using rational exponents, we need to understand that taking the sixth root of a number is equivalent to raising that number to the power of \( \frac{1}{6} \).

Step 2: Expressing the Radical as a Power

We convert the radical expression to a rational exponent representation: \( 4^{\frac{1}{6}} \). In this step, we have made use of the fact that \( \sqrt[n]{a} = a^{\frac{1}{n}} \). Here, our radical base is 4 and the index is 6.

Key Concepts

Radicals

Radicals, commonly known as root expressions, are operations that involve finding the root of a number. In arithmetic and algebra, radical signs (like the square root symbol \( \sqrt{} \)) are used to denote roots. For example, \( \sqrt{4} \) denotes the square root of 4, which is 2. The number inside the radical sign is called the radicand.
In expressions involving radicals, the index of the radical tells you which root to take. If no index is written, it's assumed to be a square root (index 2). So, \( \sqrt[3]{8} \) means the cube root of 8. For the exercise \( \sqrt[6]{4} \), the index is 6, meaning we're finding the sixth root of 4.

• Radicals answer questions like "which number multiplies by itself a certain number of times to get the radicand?"
• Besides square roots, there are cube roots, fourth roots, and so on, each with their respective indices.
• The process of converting radicals to rational exponents makes it often easier to manipulate radical expressions in algebraic problems.

Simplifying Expressions

Simplifying expressions is about reducing expressions to their most basic form. This involves using various algebraic methods to make them easier to understand and work with.
In the context of radicals, simplifying means rewriting the expression in a way that makes calculations easier or reveals certain properties of the expression.
For example, the sixth root of 4, \( \sqrt[6]{4} \), can be simplified by converting it into a rational exponent. This translates to \( 4^{\frac{1}{6}} \), as demonstrated in the original solution. Using rational exponents lets us apply laws of exponents more easily:

• Product of powers: \( a^m \times a^n = a^{m+n} \)
• Quotient of powers: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a power: \( (a^m)^n = a^{m\cdot n} \)

This particular simplification allows for deeper mathematical operations and ensures expressions remain consistent and straightforward.

Real Numbers

Real numbers are the set of numbers that includes all the numbers you can find on the number line. This includes all rational and irrational numbers.
Rational numbers are numbers that can be expressed as the ratio of two integers, like \( \frac{1}{2} \) or 3. Irrational numbers, on the other hand, cannot be expressed as simple fractions, like \( \pi \) or \( \sqrt{2} \).
In working with radicals and expressions involving rational exponents, it's essential to understand their domain often lies within the set of real numbers.

• Radicals with even indices (like square roots) applied to negative numbers do not result in real numbers, but rather imaginary numbers.
• Real numbers are seamless and continuous across the number line, which is why they work harmoniously with radicals and rational exponents.

When simplifying the expression \( \sqrt[6]{4} \), the number 4 is assumed to be a positive real number, allowing us to fairly and confidently work with its radical and exponent forms.