Problem 15 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Express 45 as a Product of Its Prime Factors

First, express 45 as a product of its prime factors to simplify the square root. The number 45 can be factored into 3 and 15, where 15 can be further broken down into 3 and 5. Thus, the prime factorization of 45 is \(3 \times 3 \times 5\).

Step 2: Simplify the Square Root

Using the prime factorization, \(45 = 3^2 \times 5\). The square root of 45 becomes \(\sqrt{45} = \sqrt{3^2 \times 5}\). Apply the property of square roots that states \(\sqrt{a^2 \times b} = a \sqrt{b}\), to simplify it to \(3 \sqrt{5}\).

Step 3: Multiply by the Scalar Outside the Square Root

Since we have \(8 \sqrt{45}\), replace \(\sqrt{45}\) with the simplified square root expression from the last step. Thus, \(8 \sqrt{45} = 8 \times 3 \sqrt{5}\).

Step 4: Perform the Multiplication

Multiply the scalar 8 by 3 to simplify the expression fully: \(8 \times 3 \sqrt{5}\). This results in \(24 \sqrt{5}\).

Key Concepts

Prime Factorization

Prime factorization is like finding all the building blocks of a number. These building blocks are prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. When you factor a number into its prime components, each of these prime numbers is like an ingredient in a recipe.
For example, to find the prime factorization of 45, you begin by dividing it by the smallest prime number that fits, which is 3 in this case:

• 45 divided by 3 gives 15.
• 15 can again be divided by 3, resulting in 5.
• Finally, 5 is a prime number.

So, 45 can be represented as the product of its prime factors: \(3 \times 3 \times 5\). This process of breaking down a number into prime factors is crucial when simplifying square roots, as it helps identify perfect squares.

Square Root Properties

Square roots have special properties that allow us to simplify them effectively. One essential property is if you have a number under the square root that is a product of squares and nonsquares, you can simplify it.
Here's how it works with the example of 45, which we've already factored: \(45 = 3^2 \times 5\). When you take the square root of 45, you can first separate it into \(\sqrt{3^2} \times \sqrt{5}\). The square root of \(3^2\) is simply 3, since raising a number to the power of 2 and then taking the square root eliminates each other:

• \(\sqrt{3^2} = 3\)
• Keep \(\sqrt{5}\) as it is, because 5 is not a perfect square.

Therefore, \(\sqrt{45}\) simplifies to \(3 \sqrt{5}\). Recognizing this property allows us to simplify square roots that seem complicated at first glance.

Multiplying Radicals

When it comes to multiplying radicals, it's essential to understand how to handle numbers both inside and outside the root efficiently. If you have an expression like \(8 \sqrt{45}\), once you simplify \(\sqrt{45}\) to \(3 \sqrt{5}\), the next step is straightforward multiplication.
You multiply the number outside the square root by the number outside of any square root in the expression you're dealing with. In our problem, you have:

• 8 (outside the square root) and 3 (also outside)
• Multiply these together, giving us 24.
• Keep \(\sqrt{5}\) as it is.

So, the entire expression \(8 \times 3 \sqrt{5}\) simplifies to \(24 \sqrt{5}\). This ability to multiply elements outside the radicals makes simplifying much more manageable, helping you get through radical problems efficiently.