Problem 15 - Intermediate Algebra

Question

Simplify each expression. $$ 8 \sqrt{45} $$

Step-by-Step Solution

Answer

The simplified form is \(24 \sqrt{5}\).

Step 1: Factor the Radicand

To simplify the square root, first factor the number under the square root sign (radicand). For \(45\), the prime factorization is \(45 = 3^2 imes 5\).

Step 2: Apply the Square Root to the Factors

Next, apply the square root to each of the factors. Since \(45\) is \(3^2 imes 5\), you can separate it as \( \sqrt{3^2} imes \sqrt{5}\).

Step 3: Simplify the Square Root

Now extract the square root of the perfect square factor. \(\sqrt{3^2} = 3\), so the expression becomes \(3 \times \sqrt{5}\).

Step 4: Multiply by the Coefficient Outside the Square Root

The expression is now \( 8 \times 3 \times \sqrt{5}\), multiply the numbers outside the square root: \(8 \times 3 = 24\).

Step 5: Write the Simplified Expression

Combine the results: \(24 \sqrt{5}\). The simplified form of \(8\sqrt{45}\) is \(24\sqrt{5}\).

Key Concepts

Prime Factorization

Prime factorization is a method of breaking down a composite number into a product of prime numbers. Prime numbers are numbers greater than 1 that do not have any divisors other than 1 and themselves. For example, to determine the prime factorization of the number 45, we need to break it down into the smallest possible factors that are all prime numbers.
Let's begin by dividing 45 by the smallest prime number, 3, which gives us 15. We can further divide 15 by 3 to get 5, which is a prime number. Thus, the prime factorization of 45 can be expressed as:

• 45 = 3 x 3 x 5

This can also be written as \(3^2 \times 5\). Prime factorization helps in simplifying expressions by identifying factors that are perfect squares, making it easier to work with square roots.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. The square root is often represented by the radical symbol \(\sqrt{}\). For instance, the square root of 9 is 3 because \(3 \times 3 = 9\).
When simplifying square roots, the goal is to reduce the expression inside the radical as much as possible by factoring out perfect squares.
In our exercise, the square root of 45 can be expressed using its prime factors:

• \(\sqrt{45} = \sqrt{3^2 \times 5}\)

This allows us to simplify the expression by extracting the square root of the perfect square, \(3^2\), resulting in \(3\sqrt{5}\). Understanding how to evaluate square roots through prime factorization is crucial for simplifying radical expressions.

Radicand

The term "radicand" refers to the number or expression under the radical sign in a square root or any other root. It is the number you wish to find the root of. In the expression \(\sqrt{45}\), 45 is the radicand.
The radicand plays a key role in the simplification process because its factors determine how the expression can be broken down or simplified.
When looking to simplify \(\sqrt{45}\), you realize that factoring the radicand into its prime factors enables you to identify perfect squares. This allows you to break down the expression more easily, as seen in:

• \(\sqrt{45} = \sqrt{3^2 \times 5}\)

Identifying the radicand is the first step in simplifying square roots and other similar expressions.

Perfect Squares

A perfect square is a number that can be expressed as the product of an integer with itself. Common examples include numbers like 1, 4, 9, 16, and so on. Recognizing perfect squares is beneficial when simplifying square roots because they allow you to extract square roots more easily.
In the context of our exercise, part of simplifying \(\sqrt{45}\) involves factoring the radicand and identifying any perfect squares among its factors.

• For \(3^2\), the number 9 is a perfect square.

This squares up to 3 (\(\sqrt{9} = 3\)), allowing us to pull it out from under the radical. So, \(\sqrt{3^2}\) simplifies to 3, which reduces the expression \(\sqrt{45}\) to \(3\sqrt{5}\). Recognizing and leveraging perfect squares makes expressions less cumbersome and much simpler to work with.