Problem 23 - Algebra and Trigonometry

Question

Simplify the expression. \(\sqrt{32}+\sqrt{18}\)

Step-by-Step Solution

Answer

The expression simplifies to \(7\sqrt{2}\).

Step 1: Break Down the Radicals

Start by breaking down each square root into its prime factors. Begin with \( \sqrt{32} \). Since 32 can be factored as \( 2^5 \):\[\sqrt{32} = \sqrt{2^5} = \sqrt{(2^4) \times 2} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2}\]Since \( \sqrt{16} = 4 \), this simplifies to:\[4\sqrt{2}\]Next, simplify \( \sqrt{18} \). Since 18 can be factored as \( 2 \times 3^2 \):\[\sqrt{18} = \sqrt{2 \times 9} = \sqrt{9} \times \sqrt{2}\]Since \( \sqrt{9} = 3 \), this simplifies to:\[3\sqrt{2}\]

Step 2: Combine Like Terms

Now that we have \( \sqrt{32} = 4\sqrt{2} \) and \( \sqrt{18} = 3\sqrt{2} \), combine the like terms:\[4\sqrt{2} + 3\sqrt{2} = (4 + 3)\sqrt{2} = 7\sqrt{2}\]

Step 3: Write the Final Expression

The simplified version of the original expression \( \sqrt{32} + \sqrt{18} \) is therefore:\[7\sqrt{2}\]

Key Concepts

Prime Factorization

Prime factorization is breaking down a number into its basic building blocks, which are prime numbers. It is an essential step in simplifying radicals.

• Prime numbers are those greater than 1 that have no divisors other than 1 and themselves, such as 2, 3, 5, 7, etc.
• In the context of radicals, prime factorization helps us decompose the number inside the square root into smaller factors that are easier to manage.

When simplifying a square root, as seen in the exercise, you first take the number under the radical sign, like 32 or 18, and express them in terms of their prime factors. For 32, that is:
32 = 2 x 2 x 2 x 2 x 2 = 2⁵.
For 18, it is:
18 = 2 x 3 x 3 = 2 x 3².
This breaking down allows us to pair up factors, which can then be easily taken out of the square root when their number is even.

Like Terms

Like terms are terms that have the same variables and powers, which allows them to be combined through addition or subtraction. In the case of simplifying expressions involving radicals, like terms play a crucial role.

• Terms are considered 'like' if they have the same radical part.
• For example, in the expression from the exercise: both expressions, \(4\sqrt{2}\) and \(3\sqrt{2}\), have the same \(\sqrt{2}\) part.

This commonality allows us to add them together by combining just the numerical coefficients, giving us:
\[4\sqrt{2} + 3\sqrt{2} = (4 + 3)\sqrt{2} = 7\sqrt{2}\]Finding and combining like terms is vital in simplifying radical expressions efficiently.

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. Radicals are often represented using the square root symbol: \( \sqrt{} \). Understanding square roots is key to simplifying radical expressions like \(\sqrt{32}\) and \(\sqrt{18}\).

• To simplify a square root, such as \(\sqrt{32}\), we break down 32 into its prime factors and look for perfect square pairs.
• For instance, 32 can be written as \(16 \times 2\) where 16 is a perfect square being \((4 \times 4)\).
• Since \(\sqrt{16} = 4\), it can be taken out of the radical, simplifying \(\sqrt{32}\) to \(4\sqrt{2}\).

The same methodology applies to \(\sqrt{18}\) where breaking it into factors, \(9\times 2\), allows you to simplify it as \(3\sqrt{2}\) since \(\sqrt{9} = 3\). Understanding square roots empowers you to simplify complex radical expressions effectively.