Problem 1 - Algebra for College Students

Question

Use the distributive property to help simplify each of the following. \(5 \sqrt{18}-2 \sqrt{2}\)

Step-by-Step Solution

Answer

The expression simplifies to \(13 \sqrt{2}\).

Step 1: Identify the Radicals

Examine each term and identify any radicals that can be simplified. We have \( \sqrt{18} \) and \( \sqrt{2} \).

Step 2: Simplify Radicals

Simplify \( \sqrt{18} \) by finding the prime factorization: \( \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3 \sqrt{2} \).

Step 3: Apply the Distributive Property

Substitute the simplified form of \( \sqrt{18} \) into the expression: \( 5 \times 3 \sqrt{2} - 2 \sqrt{2} \). This becomes \( 15 \sqrt{2} - 2 \sqrt{2} \).

Step 4: Combine Like Terms

Combine the terms with \( \sqrt{2} \) as the common factor: \( (15-2) \sqrt{2} = 13 \sqrt{2} \).

Key Concepts

Radical Simplification

Radical simplification involves breaking down a radical expression into its simplest form. When dealing with radicals, it helps to simplify them to work with easier numbers. Consider the expression \( \sqrt{18} \). To simplify it, you start by performing prime factorization:

• Break down the number 18 into its prime factors. We find that 18 can be factored as 9 and 2, where 9 is a perfect square \((3^2)\).
• Write \( \sqrt{18} = \sqrt{9 \times 2} \), which can be separated into \( \sqrt{9} \times \sqrt{2} \).
• Further simplify \( \sqrt{9} \) to 3, because \( 3^2 = 9 \).

In this way, \( \sqrt{18} \) simplifies to \( 3 \sqrt{2} \). Recognizing and simplifying radicals makes it much easier to combine and manipulate them in expressions.

Prime Factorization

Prime factorization is the process of breaking a number down into its basic building blocks, or prime numbers. It's an essential tool for radical simplification. Let's consider how it was applied to \( \sqrt{18} \):

• A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
• Prime factorization of 18 is done by dividing it into the smallest prime numbers: 18 can be expressed as \( 2 \times 3^2 \).
• This means \( \sqrt{18} \) can be rewritten to \( \sqrt{9 \times 2} \) simplifying it to \( 3 \sqrt{2} \).

Understanding prime factorization is key to reducing complex radicals into simpler forms, facilitating easier calculations and manipulations.

Combining Like Terms

Combining like terms is a fundamental algebraic process that simplifies expressions, especially after distributing. In the given problem, we've gotten to a point where we have:

• Terms \( 15 \sqrt{2} - 2 \sqrt{2} \) present, which both include the like term \( \sqrt{2} \).
• Combine them by performing basic arithmetic operations on their coefficients: \( 15 - 2 = 13 \).
• This results in the combined expression: \( 13 \sqrt{2} \).

Creating unified terms from similar ones not only simplifies expressions but also makes further calculations and understanding much easier. Recognizing and combining these like terms is crucial for completing algebraic problems efficiently.