Problem 27 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{18} \cdot \sqrt{5} $$

Step-by-Step Solution

Answer

The simplified expression of the given problem is equal to \(3\sqrt{10}\).

Step 1: Identifying square roots

First we need to identify the square roots in the expression. The square root signs indicate that there are two square roots that need to be calculated, which are \( \sqrt{18} \) and \( \sqrt{5} \).

Step 2: Simplify the square roots

Then, we simplify the square roots. Start by simplifying \( \sqrt{18} \). The prime factorization of 18 is \( 2 \times 3^2 \). Therefore, \( \sqrt{18} \) can be rewritten as \( \sqrt{2 \times 3^2} \), and simplifies to \( 3 \sqrt{2} \). The second square root, \( \sqrt{5} \), is already in its simplest form because 5 is a prime number.

Step 3: Multiply the Resulting Expressions

Finally, multiply the two simplified expressions together: \( (3 \sqrt{2}) \times \sqrt{5} \). This is equals to \( 3 \sqrt{2} \times \sqrt{5} \), which further simplifies to \( 3 \sqrt{10}. \)

Key Concepts

Prime Factorization

Prime factorization is like finding the smallest building blocks that make up a number, using only prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves, like 2, 3, 5, 7, etc.
To factor a number, break it down until all you have are prime numbers. For example, the number 18 can be broken down as follows:

• Divide 18 by 2 (the smallest prime number and a factor of 18) to get 9.
• Divide 9 by 3, another prime number, to get 3.
• Divide 3 by 3 to finally get 1, completing the factorization.

Thus, the prime factorization of 18 is \( 2 \times 3^2 \). We use this factorization to simplify square roots, because pairs of identical factors can "come out" of the square root.

Multiplying Radicals

When multiplying radicals, like square roots, you can often simplify the expression by combining what's under the root signs before simplifying further. The product under a single square root is the same as the two separate products outside the roots.
This means if you have \( \sqrt{a} \times \sqrt{b} \), you can rewrite it as \( \sqrt{a \times b} \). For instance, with \( \sqrt{18} \cdot \sqrt{5} \), you can combine them to get \( \sqrt{18 \times 5} = \sqrt{90} \).
A simple way to handle this is:

• Simplify each square root if possible (like \( \sqrt{18} \) to \( 3\sqrt{2} \)).
• Multiply only the numbers outside the radicals together.
• Perform the same multiplication under the square root and simplify further if needed.

In our example, \( \sqrt{90} \) can be simplified if desired, but the given exercise already stops at \( 3 \sqrt{10} \).

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the symbol \( \sqrt{} \).
The square root simplifies numbers by pulling out perfect squares. For instance, the square root of 9 is 3, since \( 3 \times 3 = 9 \). While numbers like 5, without perfect squares, remain unchanged unless expressed in terms of decimals.
This makes understanding concepts like \( \sqrt{18} \) crucial when simplifying. By using prime factorization, notice \( \sqrt{2 \times 3^2} \) simplifies as 3 escapes the root, being a pair, leading to \( 3 \sqrt{2} \).
Key steps include:

• Identify pairs in prime factors to "take out" from under the radical.
• Keep terms without pairs under the root, simplifying where possible.

This process makes expressions less complex and easier to handle, especially when multiplying.