Problem 44 - College Algebra

Question

Simplify each expression. $$3 \sqrt{44 z}+\sqrt{99 z}$$

Step-by-Step Solution

Answer

The simplified expression is \( 9\sqrt{11z} \).

Step 1: Simplify the Square Roots

First, we need to simplify each square root separately. To simplify \( \sqrt{44z} \), we find the largest perfect square factor of 44, which is 4. So, \( \sqrt{44z} = \sqrt{4 \times 11z} = \sqrt{4} \sqrt{11z} = 2\sqrt{11z} \). Similarly, for \( \sqrt{99z} \), the largest perfect square factor of 99 is 9. Therefore, \( \sqrt{99z} = \sqrt{9 \times 11z} = \sqrt{9} \sqrt{11z} = 3\sqrt{11z} \).

Step 2: Combine Like Terms

The expressions \( 3 \sqrt{44z} \) and \( \sqrt{99z} \) are now simplified to \( 3(2\sqrt{11z}) \) and \( 3\sqrt{11z} \) respectively. This means the equation becomes \( 6\sqrt{11z} + 3\sqrt{11z} \).

Step 3: Add the Coefficients

Since both terms have the same radical part \( \sqrt{11z} \), we can add their coefficients. Thus, \( 6\sqrt{11z} + 3\sqrt{11z} = (6 + 3)\sqrt{11z} = 9\sqrt{11z} \).

Key Concepts

Perfect Square Factor

When simplifying radicals, one of the key steps involves identifying the greatest perfect square factor of a number under the square root. Think of perfect squares as numbers like 1, 4, 9, 16, and so on. These numbers are the result of squaring whole numbers (e.g., 2 squared is 4, 3 squared is 9).

To simplify a radical, let's use the example in the exercise: \(\sqrt{44z}\). We first find the perfect square factor of 44, which is 4 because 4 times 11 equals 44 and 4 is a perfect square (since \(2^2 = 4\)). Once identified, we can break down the square root: \(\sqrt{44z} = \sqrt{4 \times 11z} = \sqrt{4} \sqrt{11z}\). This translates to \(2\sqrt{11z}\) as \(\sqrt{4} = 2\).

Similarly, with \(\sqrt{99z}\), the largest perfect square factor is 9. By rewriting it as \(\sqrt{9 \times 11z}\), it becomes \(3\sqrt{11z}\) because \(\sqrt{9} = 3\). This process of identifying perfect square factors is crucial as it greatly simplifies the radical expressions before combining them.

Combining Like Terms

After simplifying individual square roots in an expression, the next step is to combine like terms. Like terms are terms in the expression that have the same radical part. Think of the radical part like a common variable, just as you would in the expression \( \text{3a} + \text{5a} \).

For the provided exercise, after simplifying, we have \(3(2\sqrt{11z})\) and \(3\sqrt{11z}\), which simplifies further into \(6\sqrt{11z}\) and \(3\sqrt{11z}\). Notice how both terms have the same \(\sqrt{11z}\) part. This commonality allows us to then add the coefficients (the numbers in front of the square root) together, much like adding 3a + 5a to get 8a.

So, combining \(6\sqrt{11z}\) and \(3\sqrt{11z}\) will give us \((6 + 3)\sqrt{11z} = 9\sqrt{11z}\). This simplification step streamlines the equation, making it much easier to manage and understand.

Square Root Properties

The properties of square roots are fundamental when tackling expressions involving radicals. A primary property is that the square root of a product equals the product of the square roots of each factor. In math terms, \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\). This is the heart of simplifying square root expressions.

Let's dive into an example: with \(\sqrt{44z}\) simplified as \(\sqrt{4 \times 11z}\), we can split this into \(\sqrt{4} \times \sqrt{11z}\) because of this property. This results in \(2\sqrt{11z}\). Similarly, using this property on \(\sqrt{99z} = \sqrt{9 \times 11z}\) yields \(3\sqrt{11z}\).

Additionally, to combine these expressions correctly, another property is that terms must have the same square root part to be combined directly. This ensures coherency in simplifying larger expressions and preventing mistakes. Understanding and applying these properties make the process of simplifying and combining radicals systematic and straightforward.