Problem 9 - Algebra 2

Question

Simplify. $$ \sqrt{18}+\sqrt{32} $$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{18} + \sqrt{32}\) is \(7\sqrt{2}\).

Step 1: Prime Factorization of 18

Start by finding the prime factors of 18. The prime factors of 18 are 2 and 3, in the form \(2 \times 3^2\). So, \(\sqrt{18}\) can be broken down to \(\sqrt{2 \times 3^2}\).

Step 2: Simplifying the Square Root of 18

Simplify \(\sqrt{2 \times 3^2}\) to \(3\sqrt{2}\). Here, this happens because the square root of \(3^2\) is 3, which can be taken out of the square root.

Step 3: Prime Factorization of 32

Next, find the prime factors of 32. The prime factors of 32 are 2, in the form \(2^5\). So, \(\sqrt{32}\) can be broken down to \(\sqrt{2^5}\).

Step 4: Simplifying the Square Root of 32

Simplify \(\sqrt{2^5}\) to \(4\sqrt{2}\). Here, since the square root of \(2^4\) is 4, which can be taken out of the square root.

Step 5: Combine and Simplify

Last step is to combine the simplified square roots: \(3\sqrt{2} + 4\sqrt{2} = 7\sqrt{2}\). Here, the bases are the same, so they can be added together.

Key Concepts

Prime Factorization

Prime factorization is a method used to break down a number into its basic building blocks, which are prime numbers. To find the prime factors of a number like 18, you need to divide it by the smallest prime number, which is 2, and keep dividing until you reach a result that is a prime number.
For 18, the process is:

• 18 divided by 2 equals 9
• 9 is not divisible by 2, so try the next prime number, 3
• 9 divided by 3 equals 3 (3 is a prime number)

So, the prime factorization of 18 is written as \(2 \times 3^2\).
Similarly, for 32, continue dividing by 2 as it remains divisible:

• 32 divided by 2 equals 16
• 16 divided by 2 equals 8
• 8 divided by 2 equals 4
• 4 divided by 2 equals 2
• 2 is a prime number

Thus, the prime factorization of 32 is \(2^5\). Prime factorization is critical as it lays the foundation for simplifying square roots.

Square Roots

Simplifying square roots is a process that often involves using the results of prime factorization.When you break a number into its prime components, you can look for groups of identical factors.For square roots, you're interested in pairs, as the square root of a number squared is the number itself.
Consider the example of \(\sqrt{18}\). From its prime factorization \(2 \times 3^2\), notice that 3 appears in a pair:

• The square root of \(3^2\) is 3, which can be taken out from under the root.

This simplifies \(\sqrt{18}\) to \(3\sqrt{2}\), leaving 2 inside the root as it's not paired.
Similarly, for \(\sqrt{32}\) which is expressed as \(\sqrt{2^5}\), look for pairs:

• Two pairs of 2 (\(2^4\)) can come out as 4, simplifying it to \(4\sqrt{2}\).

The remaining 2 stays inside the square root.
This technique simplifies calculations and make expressions like these easier to work with.

Combining Like Terms

Combining like terms is a critical skill in algebra that simplifies expressions and makes calculations manageable. When you have terms like \(3\sqrt{2}\) and \(4\sqrt{2}\), they are "like terms" because both involve the same square root base, \(\sqrt{2}\).
Just like adding apples to apples, you can add or subtract these terms directly:

• Add their coefficients: 3 + 4 = 7

Thus, \(3\sqrt{2} + 4\sqrt{2} = 7\sqrt{2}\).
The key is that we can only combine terms whose square root components are identical. If there were different roots, such as \(\sqrt{3}\), they wouldn't be directly combinable.This concept streamlines the process of working with complex algebraic expressions by focusing on simplifying to the most essential terms.