Problem 39 - Algebra and Trigonometry

Question

\(39-48=\) Simplify the expression. $$ \sqrt{32}+\sqrt{18} $$

Step-by-Step Solution

Answer

\( 7\sqrt{2} \)

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

First, we need to simplify \( \sqrt{32} \). We can break down 32 into its prime factors: \( 32 = 2^5 \). So, \( \sqrt{32} = \sqrt{2^5} = \sqrt{2^4 \cdot 2} = \sqrt{(2^2)^2 \cdot 2} \). This simplifies to \( 2^2 \cdot \sqrt{2} = 4\sqrt{2} \).

Step 2: Simplify \( \sqrt{18} \)

Now, simplify \( \sqrt{18} \). The number 18 can be expressed as a product of its prime factors: \( 18 = 2 \cdot 3^2 \). So, \( \sqrt{18} = \sqrt{2 \cdot 3^2} = \sqrt{3^2 \cdot 2} = 3\sqrt{2} \).

Step 3: Add \( 4\sqrt{2} + 3\sqrt{2} \)

Now, add the simplified square roots from the previous steps: \( 4\sqrt{2} + 3\sqrt{2} \). Since these are like terms (both have \( \sqrt{2} \)), we can combine them to get \( (4 + 3)\sqrt{2} = 7\sqrt{2} \).

Key Concepts

Prime Factorization

Prime factorization is about breaking down numbers into their most basic building blocks: prime numbers. Prime numbers are the numbers that are greater than 1 and cannot be formed by multiplying two smaller numbers. The process is simple.
While working with square roots, prime factorization helps in simplifying such expressions. For example:

• To factor 32, we need to find numbers that multiply together to return 32, using primes. Here, 32 can be factored as \( 32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5 \).
• In this breakdown, 2 is a prime number, and it's multiplied by itself five times.

Similarly, when dealing with 18, the prime factors are \( 2 \) and \( 3^2 \). Each factor contributes to simplifying the radical expression further by grouping the factors. This step is crucial because it sets the stage for the next processes, like simplifying square roots.

Like Terms

In algebra, the concept of like terms is hugely important. Like terms have the same variables raised to the same powers. They can be combined with one another.
In the context of radical expressions, terms are like terms if they have the same radical part. Now, what does that mean?

• Consider \( 4\sqrt{2} \) and \( 3\sqrt{2} \). The \( \sqrt{2} \) part is what makes these terms alike.
• Because their radical parts are the same, you can add the coefficients (4 and 3 in this example) together.

So, when you simplify \( 4\sqrt{2} + 3\sqrt{2} \), you simply add 4 and 3, giving \( 7\sqrt{2} \). Figuring out which terms are alike is essential when simplifying or combining expressions in algebra.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. Simplifying these expressions is necessary for making calculations easier and more manageable.
Let's break down what it means to simplify a radical expression.

• Firstly, you use prime factorization to find any doubles inside the radical sign. These doubles are your key to simplifying.
• For example, in \( \sqrt{32} = \sqrt{2^5} \), you can pair four 2's as two sets of 2 (\((2^2)^2 \)), and bring them out as 4 (since you take one 2 from each pair out of the square root).
• Once you've simplified each square root separately, you might encounter like terms, as explained before. These can then be combined for an even simpler answer.

Therefore, understanding radical expressions and how to simplify them is fundamental in algebra to solve equations effectively. Simple steps, like finding pairs under the root or knowing when two radicals are similar, empower you to simplify even complex expressions.