Problem 19 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{32}+\sqrt{2} $$

Step-by-Step Solution

Answer

The simplified form of the given expression \(\sqrt{32} + \sqrt{2}\) is \(5\sqrt{2}\)

Step 1: Simplify \(\sqrt{32}\)

In this step, we simplify \(\sqrt{32}\). We can breakdown 32 into a product of 16 and 2, and we know that the square root of 16 is 4. Therefore, \(\sqrt{32} = \sqrt{16}\times \sqrt{2} = 4\sqrt{2}\)

Step 2: Simplify Given Expression

Now, we substitute the simplified form of \(\sqrt{32}\) into the initial expression. Therefore, the expression becomes \(4\sqrt{2} + \sqrt{2}\)

Step 3: Perform Addition

Now, we perform the addition. The like terms \(4\sqrt{2}\) and \(\sqrt{2}\) can be added together to get \(5\sqrt{2}\).

Key Concepts

Square Roots

A square root is a number that produces a specified quantity when multiplied by itself. For example, the square root of 16 is 4 because when you multiply 4 by itself (4 x 4), you get 16. Understanding square roots is essential, especially when simplifying expressions that involve them. To simplify roots, look for perfect square factors.

Let's break down the original expression:

• The expression \(\sqrt{32}\) can be thought of as \(\sqrt{16 \times 2}\).
• Since 16 is a perfect square, it's easy to simplify: \(\sqrt{16} = 4\).

Thus, \(\sqrt{32}\) simplifies to \(4 \sqrt{2}\). Recognizing these perfect squares is key when simplifying complex square roots.

Addition of Like Terms

When you're adding algebraic expressions, you often encounter like terms. Like terms are terms that have the same variable raised to the same power. It's crucial to understand this concept, especially when working with radicals like square roots.

In the expression \(4\sqrt{2} + \sqrt{2}\), both terms are 'like' because they involve \(\sqrt{2}\).

• You treat the square root part as a common variable or label, and simply add the numerical coefficients.
• So, \(4\sqrt{2} + 1\sqrt{2} = (4+1)\sqrt{2} = 5\sqrt{2}\).

This is similar to adding \(4x + 1x\) to get \(5x\). Always combine like terms to simplify your expressions.

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, which are prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.

So how does this relate to simplifying square roots?

• By identifying the prime factors of a number, you can quickly find which parts are perfect squares.
• For instance, we factor 32 into prime numbers: \(32 = 2 \times 16 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5\).

Knowing this helps identify perfect squares (like 16 in \(\sqrt{32}\)), making it simpler to simplify the radical further. Using prime factorization, you can ensure all square roots are expressed in their simplest form for ease of addition or multiplication.