Question

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$\sqrt{18}$$

Short answer

The simplest form of \(\sqrt{18}\) is \(3\sqrt{2}\).

Step-by-step solution

Factor the Radicand

Start by factoring the number 18 into its prime factors: 18 = 2 × 3 × 3.

Simplify the Square Root

Identify pairs of prime factors to simplify the square root. Since we have a pair of 3's, we can take one 3 out of the square root, leaving the 2 inside:\[\sqrt{18} = \sqrt{2 × 3²} = 3\sqrt{2}\]

Write the Final Answer

The simplest radical form of \(\sqrt{18}\) is \(3\sqrt{2}\).

Key concepts

Radical Form

When dealing with square roots and other radical expressions, understanding what 'radical form' means is crucial for simplifying these expressions correctly. Radical form is a way of expressing numbers that includes a radical sign \( \sqrt{} \) and indicates the root of a number. For square roots, which are the most common form of radicals, the radical sign without an index (or with an index of 2) implies that we are finding the number that, when multiplied by itself, gives us the radicand—the number under the radical sign.

For example, the radical form of 18 is expressed as \( \sqrt{18} \) because we're looking for the number that, when squared, equals 18. The aim of simplification is to find the most straightforward expression for the square root, ideally with no perfect square factors remaining in the radicand. This process often involves identifying and removing pairs of prime factors, which leads us to our next concept: prime factorization.

Prime Factorization

Prime factorization involves breaking down a composite number into its prime factors. Prime factors are the building blocks of numbers—they are the prime numbers that multiply together to result in the original number, and every composite number has a unique set of prime factors.

To perform prime factorization, we repeatedly divide the number by the smallest possible prime numbers until we're left only with prime numbers. With the number 18, we start by recognizing that it's an even number, so 2 is a prime factor. 18 divided by 2 gives us 9, which is not prime, but can be expressed as 3 times 3—both prime numbers. So, the prime factorization of 18 is 2 × 3 × 3, which is also written as 2 × 3² to denote that 3 is a factor twice.

• Prime Factorization of 18: 2 × 3 × 3

Prime factorization is essential when simplifying square roots because it helps us to identify pairs of prime factors that can be taken out of the radical sign, thereby simplifying the radical expression.

Square Root Simplification

Simplifying square roots is about writing the square root of a number in its simplest radical form. This process removes perfect square factors from underneath the radical sign to outside, where they can be represented as their square root.

The general steps to simplify square roots are:

1. Perform prime factorization on the radicand.
2. Identify pairs of equal prime factors.
3. Bring one number from each pair out in front of the radical sign and express it without the radical.
4. Multiply the numbers outside the radical together (if more than one), and do the same with any remaining factors inside the radical.

For our example, the square root of 18 is simplified as follows:

• \( \sqrt{18} = \sqrt{2 \times 3^2} \) (after prime factorization)
• A pair of 3's is identified.
• The pair of 3's comes out in front of the radical as a single 3.
• The prime number 2 remains inside the radical.

Thus, the simplified form is \( 3\sqrt{2} \)—this is the simplest way to express \( \sqrt{18} \) in radical form. Knowing how to simplify square roots is valuable for algebra, geometry, and higher mathematics. It also helps when dealing with quadratic equations and understanding the properties of numbers.