Problem 83 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ 2 \sqrt{3 z}+3 \sqrt{12 z}+3 \sqrt{48 z} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 20 \sqrt{3z} \).

Step 1: Simplify Each Square Root Term

First, simplify each square root in the expression individually. - The first term is already in its simplest form: \( 2 \sqrt{3z} \).- For the second term, \( 3 \sqrt{12z} \), simplify \( \sqrt{12} \) to \( \sqrt{4 \times 3} = 2 \sqrt{3} \), resulting in the term: \( 3 \times 2 \sqrt{3z} = 6 \sqrt{3z} \).- For the third term, \( 3 \sqrt{48z} \), simplify \( \sqrt{48} \) to \( \sqrt{16 \times 3} = 4 \sqrt{3} \), resulting in the term: \( 3 \times 4 \sqrt{3z} = 12 \sqrt{3z} \).

Step 2: Combine Like Terms

Now that all terms are simplified, combine them by adding the coefficients of the like terms. All terms contain the common term \( \sqrt{3z} \):- Combine \( 2 \sqrt{3z} + 6 \sqrt{3z} + 12 \sqrt{3z} = (2 + 6 + 12) \sqrt{3z} = 20 \sqrt{3z} \).

Key Concepts

Simplifying Expressions

In algebra, simplifying expressions is an essential skill that helps streamline more complex calculations by reducing expressions to their simplest form. It involves breaking down parts of the expression, such as square roots and polynomials, to make them more manageable. When you see an expression like \(2 \sqrt{3z} + 3 \sqrt{12z} + 3 \sqrt{48z}\), the goal is to transform it into a version that's easy to work with and understand.

To simplify this particular expression, we begin by focusing on each component separately. By breaking down complex parts like \(\sqrt{12}\) and \(\sqrt{48}\), we find the simplest representation possible. Simplifying not only makes calculations more straightforward but also reveals the foundational structure of the mathematical problem you're dealing with.

Adopting these simplification strategies ensures that problems are less cumbersome to solve and allows for greater focus on understanding the inherent mathematical relationships.

Square Roots

Square roots appear frequently in algebraic expressions, and understanding how to simplify them is crucial for engaging with complex algebraic forms. A square root, denoted by the symbol \(\sqrt{}\), essentially asks, "what number, when multiplied by itself, equals this number?" For instance, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).

When simplifying square roots like \(\sqrt{12}\) or \(\sqrt{48}\), factor the number under the square root symbol into its prime factors. Take \(\sqrt{12}\) as an example. You can break it down to \(\sqrt{4 \times 3}\), which simplifies further to \(2 \sqrt{3}\) since \(\sqrt{4} = 2\). Similarly, \(\sqrt{48}\) is \(\sqrt{16 \times 3}\), simplifying to \(4 \sqrt{3}\).

This process involves identifying perfect squares (like 4 and 16) within the original number, as these simplify directly to integer values, making the overall term less complex. Understanding this method simplifies handling large numbers and complicated expressions.

Combining Like Terms

Once you’ve simplified each term individually, combining like terms is your next step in taming an expression. This process focuses on consolidating similar terms, in particular those that share the same base variable and power.

In the expression \(2 \sqrt{3z} + 6 \sqrt{3z} + 12 \sqrt{3z}\), each term includes the common base \(\sqrt{3z}\). Combining these like terms involves adding their coefficients. Here, you would add 2, 6, and 12 to get 20, resulting in a single term: \(20 \sqrt{3z}\).

This technique dramatically reduces the complexity of an expression. By collapsing terms that mirror each other in structure, you streamline calculations and gain clearer insights into the problem’s nature. Mastering this concept means you'll be more adept in tackling extensive algebraic problems with ease.