Problem 44 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(3 \sqrt{44 z}+\sqrt{99 z}\)

Step-by-Step Solution

Answer

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

Step 1: Factor the Radicals

First, we need to simplify the radicals by factoring out any perfect squares. For the term \( \sqrt{44z} \), notice that \( 44 = 4 \times 11 \). Since 4 is a perfect square, we can rewrite \( \sqrt{44z} \) as \( \sqrt{4 \cdot 11 \cdot z} = \sqrt{4} \cdot \sqrt{11z} = 2\sqrt{11z} \).

Step 2: Simplify the Second Radical

Now consider \( \sqrt{99z} \). We factor \( 99 = 9 \times 11 \). Since 9 is a perfect square, we rewrite \( \sqrt{99z} \) as \( \sqrt{9 \cdot 11 \cdot z} = \sqrt{9} \cdot \sqrt{11z} = 3\sqrt{11z} \).

Step 3: Combine Like Terms

We have simplified the expression to \(3(2\sqrt{11z}) + 3\sqrt{11z}\). This becomes \(6\sqrt{11z} + 3\sqrt{11z}\) since \( 3 \cdot 2\sqrt{11z} = 6\sqrt{11z} \).

Step 4: Simplify the Expression

Combine the like terms \(6\sqrt{11z} + 3\sqrt{11z}\) to get \(9\sqrt{11z}\).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators. They don't include an equals sign, which means they are not complete equations.
Examples of algebraic expressions include phrases like \( 3x + 2 \), \( 4a^2 - 9b \), or more complex forms involving roots like \( 3 \sqrt{44z} + \sqrt{99z} \).
In algebraic expressions, we often simplify or evaluate them by following operations and combining like terms.

• **Operations:** This involves addition, subtraction, multiplication, and division.
• **Simplifying:** This is the process of reducing the expression to its simplest form. It often involves factoring, distributing terms, or combining like terms.

The main goal is to make expressions easier to understand or solve, which is why knowing how to manipulate them using algebraic rules is important.

Perfect Squares

Perfect squares are numbers that have exact square roots, meaning they can be expressed as the square of an integer.
Examples include numbers like 1, 4, 9, 16, 25, where their square roots are 1, 2, 3, 4, 5 respectively. Recognizing perfect squares is crucial when simplifying expressions involving square roots.
In the simplification process, spotting a perfect square within a radical helps us to reduce the expression easily. For instance, in the radical \( \sqrt{44z} \), we identify 4 as a perfect square and rewrite it as \( \sqrt{4} \times \sqrt{11z} = 2\sqrt{11z} \). Similarly, for \( \sqrt{99z} \), we use 9, another perfect square, turning the expression into \( 3\sqrt{11z} \).

• Perfect squares simplify square roots significantly.
• They enable us to reduce expressions, leading to a cleaner and more understandable form.

Spotting these squares requires practice but greatly aids in algebraic simplification.

Like Terms

Like terms are terms within an algebraic expression that have identical variable parts and, therefore, can be combined.
This means they must have the same variables raised to the same powers. For example, terms like \( 2x \) and \( 3x \) are like terms because they both contain the variable \( x \) to the first power.
In expressions involving radicals, like terms are those with the same radical components. In our example, both \( 6\sqrt{11z} \) and \( 3\sqrt{11z} \) share the \( \sqrt{11z} \) part, allowing us to combine them into \( 9\sqrt{11z} \).

• Combining like terms simplifies expressions.
• It makes it easier to perform further algebraic operations such as solving or factoring.

Understanding how to identify and combine like terms is a fundamental skill in algebra, helping maintain expression simplicity and clarity.