Problem 33 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt[4]{25 z} \cdot \sqrt[4]{25 z} $$

Step-by-Step Solution

Answer

The simplified expression is \( 5\sqrt{z} \).

Step 1: Identify the Problem

The given expression is \( \sqrt[4]{25z} \cdot \sqrt[4]{25z} \). We need to simplify this expression assuming all variables, including \( z \), are positive.

Step 2: Apply the Product of Roots Rule

When multiplying roots with the same degree, you can combine them under a single root: \( \sqrt[4]{a} \cdot \sqrt[4]{b} = \sqrt[4]{a \cdot b} \). Here, combine the terms under a single fourth root: \( \sqrt[4]{25z \times 25z} \).

Step 3: Simplify the Product Inside the Root

Multiply the terms inside the root: \( 25z \times 25z = 625z^2 \). Therefore, the expression becomes \( \sqrt[4]{625z^2} \).

Step 4: Factor the Expression Under the Root for Further Simplification

We rewrite \( 625 \) as \( 25^2 \) and note that \( z^2 \) is a perfect square: \( \sqrt[4]{(25^2)(z^2)} \). The expression can be simplified using the property of roots: \( \sqrt[4]{a^4} = a \).

Step 5: Extract Terms From Under the Root

Apply the property \( \sqrt[4]{a^4} = a \): \( \sqrt[4]{(25^2)} = 25^{1/2} = 5 \) and \( \sqrt[4]{z^2} = z^{2/4} = z^{1/2} = \sqrt{z} \). Thus, the entire expression becomes \( 5 \sqrt{z} \).

Key Concepts

Product of Roots Rule

When simplifying expressions that involve roots, it's essential to understand how to work with the Product of Roots Rule. This rule states that when you are multiplying two roots of the same degree, you can combine them into a single root. For example, if you have:

• \(\sqrt[n]{a} \cdot \sqrt[n]{b}\)

You can rewrite this as:

• \(\sqrt[n]{a \cdot b}\)

This rule is very helpful because it allows us to simplify expressions by reducing the number of roots we are dealing with. In the given exercise, we applied this rule to combine \(\sqrt[4]{25z} \cdot \sqrt[4]{25z}\) into \(\sqrt[4]{25z \cdot 25z}\). This simplification is an important step in making the expression easier to work with.

Factorization

Once you've applied the Product of Roots Rule, factorizing the expression inside the root can simplify the problem even further. Factorization involves breaking down numbers or expressions into their constituent parts. It allows you to identify perfect squares, cubes, or other powers, which are easier to simplify. In the initial problem, we needed to factorize \(625z^2\), which is under the fourth root. Breaking this down, we noted that:

• \(625\) is the same as \(25^2\)
• \(z^2\) is simply \(z\) raised to the power of 2, making it a perfect square

So, the expression inside the root becomes \((25^2)(z^2)\). Recognizing these factors helps us in the next step, where we apply root properties to simplify the overall expression.

Root Properties

Root properties are mathematical tools that help us simplify expressions involving roots. These properties allow us to extract terms from under the root when they are perfect powers of the degree of the root.One key property is that if you have a root of a power, such as \(\sqrt[n]{x^n}\), it simplifies directly to \(x\). This makes calculations much simpler. In our problem, we had:

• \((25^2)(z^2)\) under a fourth root

Thus, \(\sqrt[4]{(25^2)}\) simplifies to \(25^{1/2} = 5\) because \(25 = 5^2\), and \(\sqrt[4]{z^2} = z^{2/4} = z^{1/2} = \sqrt{z}\).By applying these properties, the expression \(\sqrt[4]{625z^2}\) turns into \(5\sqrt{z}\). Understanding and applying these root properties is key to effectively simplifying complex root expressions.