Problem 43 - Elementary and 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 Radical to the Fraction

We start by applying the radical (square root) to both the numerator and the denominator of the expression separately: \[ \sqrt{\frac{z^2}{16x^2}} = \frac{\sqrt{z^2}}{\sqrt{16x^2}}. \]

Step 2: Simplify the Square Root in the Numerator

Simplify \( \sqrt{z^2} \). Since \( z \) is positive, \( \sqrt{z^2} = z \). So, the expression now looks like: \[ \frac{z}{\sqrt{16x^2}}. \]

Step 3: Simplify the Square Root in the Denominator

Now, simplify \( \sqrt{16x^2} \). Since \( x \) is positive, \( \sqrt{16x^2} \) becomes \( 4x \) because \( \sqrt{16} = 4 \) and \( \sqrt{x^2} = x \). Thus, the expression is now \[ \frac{z}{4x}. \]

Step 4: Verify the Simplification

Check the simplified expression to ensure it is accurate: \[ \frac{z}{4x}. \] Both square roots have been correctly simplified, validating the expression.

Key Concepts

Simplifying Radicals

Understanding how to simplify radicals can make complex mathematical expressions more manageable. When you're simplifying radical expressions, you're essentially finding the simplest form of the root. This process often involves separating the numbers under the root sign and simplifying them step by step.

For instance, when you see a radical like \( \sqrt{\frac{z^2}{16x^2}} \), you're tasked with making this as straightforward as possible. You apply the square root to the numerator and the denominator separately. This turns the expression into \( \frac{\sqrt{z^2}}{\sqrt{16x^2}} \). From there, it's a matter of simplifying each part.

• Look for perfect squares: If a part under the square root is a perfect square, simplify it to its root.
• Apply root properties: For variables, use the properties of exponents. \( \sqrt{z^2} \) becomes \( z \) because squaring and square rooting are opposite operations.
• Maintain positivity: Remember that variables represent positive real numbers, ensuring that each simplification remains valid.

By breaking down the process into smaller steps, you make simplifying radicals less daunting.

Square Root

The square root of a number (\( \sqrt{x} \)) is a value that, when multiplied by itself, gives the original number \( x \). This is fundamental when dealing with expressions like the one in our example.

Let's zoom into how we handle square roots in different scenarios:

• Perfect Square: If the number is a perfect square, the square root is an integer. For example, \( \sqrt{16} = 4 \), as seen in simplifying \( \sqrt{16x^2} \).
• Variables with even exponents: When dealing with algebraic expressions like \( \sqrt{z^2} \), try to transform these to the variable itself. Here, \( \sqrt{z^2} = z \), assuming \( z \) is positive.
• Multiple terms: When square rooting an expression that includes variables, apply the square root to each part separately. \( \sqrt{16x^2} \) becomes \( 4x \), deconstructing into \( \sqrt{16} \times \sqrt{x^2} \).

Being familiar with these fundamentals ensures that you can navigate through potentially tricky expressions with ease.

Algebraic Fractions

An algebraic fraction is essentially a fraction where the numerator, the denominator, or both, contain algebraic expressions. Simplifying them involves understanding both the arithmetic and algebra involved.

When confronted with fractions under a square root, as in \( \sqrt{\frac{z^2}{16x^2}} \), treat the numerator and denominator separately. It simplifies the process of applying the square root.

• Separate the components: Write the fraction as the square roots of the numerator and the denominator independently, \( \frac{\sqrt{z^2}}{\sqrt{16x^2}} \).
• Simplify each part: Discover where you can reduce complexity. \( \sqrt{z^2} \) simplifies to \( z \), and \( \sqrt{16x^2} \) simplifies to \( 4x \).
• Check your work: Always make sure that the resulting fraction is in its simplest form. For this problem, it reduced neatly to \( \frac{z}{4x} \).

By tackling each component separately, you achieve a clearer understanding of the structure, making algebraic fractions less of a hurdle.