Problem 43 - Intermediate Algebra

Question

Simplify each radical expression. All variables represent positive real numbers. $$ \sqrt{\frac{z^{2}}{16 x^{2}}} $$

Step-by-Step Solution

Answer

\( \frac{z}{4x} \)

Step 1: Apply the Square Root to the Fraction

The square root of a fraction is the square root of the numerator divided by the square root of the denominator. Therefore, \( \sqrt{\frac{z^2}{16x^2}} \) simplifies to \( \frac{\sqrt{z^2}}{\sqrt{16x^2}} \).

Step 2: Simplify the Square Root of the Numerator

The expression \( \sqrt{z^2} \) simplifies to \( z \) because the square root and square cancel each other out. So, \( \sqrt{z^2} = z \).

Step 3: Simplify the Square Root of the Denominator

The expression \( \sqrt{16x^2} \) simplifies to \( 4x \). This is because \( \sqrt{16} = 4 \) and \( \sqrt{x^2} = x \). Thus, \( \sqrt{16x^2} = 4x \).

Step 4: Complete the Simplification

Combine the results from Step 2 and Step 3 to give the final simplified form: \( \frac{z}{4x} \).

Key Concepts

Understanding Square Roots

A square root is a mathematical operation that helps us find a number which, when multiplied by itself, yields the original number. When you see the square root symbol \( \sqrt{} \), it means you are looking for that special number.

Some key points to remember about square roots are:

• The square root of a perfect square like 4 is simply 2 because \( 2 \times 2 = 4 \).
• This operation can be applied to not just numbers, but also to variables, such as \( \sqrt{z^2} \) which results in \( z \), assuming \( z \) is a positive real number.
• We can use this operation on fractions too, by separately finding the square roots of the numerator and denominator.

Practicing square roots with different numbers and variables will enhance your understanding and help you simplify complex expressions with ease.

Numerator and Denominator Basics

In a fraction, such as \( \frac{z}{4x} \), the numerator is the number on top, and the denominator is the number on the bottom.
They describe the two parts of any fraction:

• The numerator indicates how many parts we have.
• The denominator shows into how many parts the whole is divided.

When simplifying radical expressions involving fractions, it is essential to apply operations to both the numerator and the denominator separately.
For example, applying a square root to a fraction means taking the square root of both the numerator and the denominator distinctly. Understanding this helps in maintaining the fraction's integrity through various mathematical operations.

Steps to Simplification

Simplification is about making an expression as straightforward as possible. This doesn't just mean removing complexity, but transforming it into its simplest form while maintaining equality.
To simplify something like \( \sqrt{\frac{z^2}{16x^2}} \), follow these steps:

• Work with the numerator and the denominator separately. For our expression, start with \( \sqrt{z^2} \), which becomes \( z \).
• Next, tackle \( \sqrt{16x^2} \) by breaking it into \( \sqrt{16} \cdot \sqrt{x^2} \), resulting in \( 4x \).
• Combine these simplified parts back into a fraction form to achieve the simplified expression \( \frac{z}{4x} \).

Through simplification, you gain a clearer and more manageable version of the original, complex expression, making further calculations or understanding easier.