Problem 28 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{5} \cdot \sqrt{8} $$

Step-by-Step Solution

Answer

Simplified form of the expression \(\sqrt{5} \cdot \sqrt{8}\) is \(2 \sqrt{10}\).

Step 1 - Multiplication of the square roots

Multiplication of \(\sqrt{5}\) and \(\sqrt{8}\) can be combined into one root, using the rule \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\). So, the expression \(\sqrt{5} \cdot \sqrt{8}\) simplifies to \(\sqrt{(5*8)}\) which equals to \(\sqrt{40}\).

Step 2 - Simplifying the square root

Next, to simplify \(\sqrt{40}\), one can break down 40 into its prime factors. The prime factorization of 40 is \(2 * 2 * 2 * 5\). For the every pair of the same number, one number can come out of the root. In this case we have one pair of 2s, so one 2 comes out. Therefore, \(\sqrt{40} = 2 \sqrt{10}\).

Key Concepts

Prime Factorization

Finding the prime factors of a number is a key step in simplifying square roots, especially when dealing with larger numbers. Prime factorization involves breaking down a number into its smallest building blocks, which are the prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves, such as 2, 3, 5, 7, 11, and so on.
To perform prime factorization, start with the smallest prime number that can divide the number without leaving a remainder. Repeat this process with the quotient until you can't factor the number any further.
For instance, to find the prime factorization of 40, you start by dividing by 2, the smallest prime:

• 40 ÷ 2 = 20
• 20 ÷ 2 = 10
• 10 ÷ 2 = 5
• 5 is a prime number, so you stop here.

Thus, the prime factorization of 40 is written as: \[40 = 2 \times 2 \times 2 \times 5\]This step is crucial as it lays the groundwork for simplifying square roots later on.

Multiplying Square Roots

The rule for multiplying square roots is simple and very useful. When you have two square roots multiplied together, such as \( \sqrt{a} \cdot \sqrt{b} \), you can combine them under a single square root: \[\sqrt{ab} \]The rule applies because of how square roots work: the square root of a product is the product of the square roots.
Here's how it works using our example of \( \sqrt{5} \cdot \sqrt{8} \):

• Use the rule to combine them: \( \sqrt{5} \cdot \sqrt{8} = \sqrt{5 \times 8} \)
• Calculate inside the root: \( 5 \times 8 = 40 \), leading to \( \sqrt{40} \)

This simplification further supports the final step, where we'll simplify \( \sqrt{40} \) using its prime factorization. Remember, always simplify inside the root first to make the calculations easier.

Simplifying Radicals

Simplifying radicals is all about reducing the expression under the square root sign to its simplest form. After using prime factorization, as we did with the number 40, identify and pair up the same prime factors. Each pair of the same number indicates one of them can "escape" or move out of the root, reducing the value inside.
Let's apply this to \( \sqrt{40} \), which we previously factored as \( 2 \times 2 \times 2 \times 5 \):

• We have a pair of 2s, allowing one 2 to "escape" the square root.
• The remaining values stay inside: \( \sqrt{10} = \sqrt{2 \times 5} \)
• Thus, \( \sqrt{40} = 2 \sqrt{10} \)

This process of simplification helps us represent the square root more clearly, making further operations—and understanding—easier. It transforms complex root expressions into simpler, more manageable parts.