Problem 85 - Algebra 1

Question

Simplify the radical expression. $$ \sqrt{396}$$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt{396} \) is \( 6\sqrt{11} \)

Step 1: Factor the number under the square root

Begin by breaking down \(396\) into its prime factors: \(2 \times 2 \times 3 \times 3 \times 11 \). This can be written as \(2^2 \times 3^2 \times 11 \).

Step 2: Simplify the radicals

Take out pairs of numbers from under the radical. A pair of numbers is a 'square' and its square root can be determined. From the factorization, you can take out \(2\) and \(3\) as square roots. The \(11\) doesn't have a pair so it stays under the square root.

Step 3: Write the simplified form

So, \( \sqrt{396} \) simplifies to \( 2*3\sqrt{11} = 6\sqrt{11} \).

Key Concepts

Prime Factorization

When simplifying radical expressions such as square roots, the process often begins with prime factorization. Prime factorization is the method of breaking down a number into its prime factors, which are numbers that are only divisible by 1 and themselves. For example, let's break down the number 396.
To perform prime factorization, start with the smallest prime number that divides the given number and continue dividing by primes until you are left with prime numbers only. In the case of 396, this would look like:

• 396 divided by 2 gives 198.
• 198 divided by 2 gives 99.
• 99 divided by 3 gives 33.
• 33 divided by 3 gives 11, which is already a prime number.

This sequence results in 396 being factorized into the prime factors of \(2 \times 2 \times 3\times 3 \times 11\)). These factors can be represented using exponent notation as \(2^2 \times 3^2 \times 11\)). Prime factorization is essential as it prepares the number for further simplification under a radical.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. It is often symbolized with a radical sign \((\sqrt{\phantom{x}})\). Say, if we want to find the square root of a perfect square like 16, the answer is 4, because \(4 \times 4 = 16\).
Finding square roots of perfect squares is straightforward, but what when the number is not a perfect square, like 396? You can still find its square root by simplifying it with prime factorization. This is where you look for pairs of the same number in the prime factors because the square root of \(a^2\) is simply \(a\). That means if a number under the radical is expressed as \(a^2 \times b\), the square root of that number is \(a\sqrt{b}\). The purpose of finding these pairs during the prime factorization stage is to take the square roots of perfect squares out of the radical, just as we would do with 396, eventually finding the square roots of its prime factors where possible.

Radical Simplification

Radical simplification involves reducing a radical expression to its simplest form. After having performed prime factorization and identified perfect squares, as with the factors of 396, you next extract pairs of prime factors from beneath the radical sign. Each pair represents the square root that can be taken out of the radical. For instance, 396 has the pairs \(2^2\) and \(3^2\).

• A pair of 2's (\(2^2\)) comes out of the radical as a single 2.
• Similarly, a pair of 3's (\(3^2\)) comes out as a single 3.

With pairs taken out, the remaining number which does not form a pair stays within the radical. In our example, the number 11 remains under the radical sign. Combining the extracted factors with the remaining radical, we get \(2\times3\sqrt{11}\), which simplifies to \(6\sqrt{11}\). This is the final simplification of the original expression \(\sqrt{396}\), demonstrating the importance of radical simplification after prime factorization.