Problem 75 - Intermediate Algebra

Question

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[4]{\frac{x^{4}}{16}} $$

Step-by-Step Solution

Answer

The simplified radical is \( \frac{x}{2} \).

Step 1: Recognize the Exponents

Identify the exponent of the numerator and denominator inside the radical. The expression inside the radical is \( \frac{x^4}{16} \). Here, the exponent of \( x \) is 4, and \( 16 = 2^4 \).

Step 2: Apply the Fourth Root Property

Use the property \( \sqrt[4]{a^4} = a \) for both the numerator and the denominator. This implies \( \sqrt[4]{x^4} = x \) and \( \sqrt[4]{16} = \sqrt[4]{2^4} = 2 \).

Step 3: Simplify the Expression

Combine the simplified terms to get \( \frac{x}{2} \). This is the result after simplifying the expression using the fourth root on both the numerator and the denominator.

Key Concepts

Exponents

Exponents are a fundamental concept in mathematics. They are used to express repeated multiplication of a number by itself. An exponent is written as a small number to the right and above the base number. For example, in the expression \( a^n \), the number \( a \) is the base, and \( n \) is the exponent. This tells us to multiply \( a \) by itself \( n \) times.
Exponents make it easier to write and work with large numbers. For instance, \( 10^3 \) represents \( 10 \times 10 \times 10 = 1000 \).

• A single number raised to an exponent is called a power, and the result depends on both the base and the exponent.
• Understanding how to manipulate exponents is crucial in simplifying expressions, especially those involving radicals.

In our exercise, recognizing that both the numerator and the denominator can be rewritten with exponents is key to simplifying the radical expression efficiently.

Fourth Root

The fourth root of a number is a special kind of radical that represents the number that, when multiplied by itself four times, gives the original number. In mathematical notation, the fourth root of \( a \) is expressed as \( \sqrt[4]{a} \).
When working with the fourth root, it's important to break down complex numbers into their prime factors as much as possible to take advantage of properties of exponents.
For example, in the expression \( \sqrt[4]{x^4} \), since the exponent inside matches the root degree, simplification is direct: \( \sqrt[4]{x^4} = x \).
The fourth root is particularly useful in our example for simplifying the fraction \( \frac{x^4}{16} \) by applying it to both the numerator and denominator, resulting in simply \( x/2 \).

Numerator and Denominator

In the context of fractions, the numerator and denominator are the two components that define the value of the fraction. The numerator is the upper part of the fraction, indicating how many parts of the whole are considered. The denominator is the lower part, which tells us the total number of equal parts the whole is divided into.
For example, in the fraction \( \frac{a}{b} \), \( a \) is the numerator, and \( b \) is the denominator.
Understanding these parts can help in simplifying fractions, especially when dealing with radicals.

• Apply the same operations to both the numerator and the denominator.
• In our example, we treat \( x^4 \) as the numerator and \( 16 \) as the denominator.
• By simplifying each part individually using the fourth root property, we arrive at an expression that is much simpler.

Real Numbers

Real numbers include all the numbers on the number line, encompassing both rational and irrational numbers. They are fundamental in mathematics, serving as a complete set for arithmetic operations.

• Positive real numbers are specifically important when dealing with radicals, as negative bases can lead to imaginary results if the root is even.
• In our problem, knowing the variables represent positive real numbers simplifies the process because we don't have to consider complex or imaginary numbers.

Real numbers are essential in ensuring the validity of expressions, such as \( \frac{x}{2} \), where \( x \) is assumed to be a positive real number, eliminating concerns over square roots of negative numbers.