Problem 46 - Algebra 1

Question

Simplify the expression. $$\sqrt{7} \cdot \frac{\sqrt{18}}{\sqrt{2}}$$

Step-by-Step Solution

Answer

\(3\sqrt{7}\)

Step 1: Break down each number under the square root

The first step is simplifying the square roots by breaking each number down into its prime factors. So, \(\sqrt{7}\) stays as \(\sqrt{7}\) (since 7 is a prime number), \(\sqrt{18}\) simplifies to \(\sqrt{2*3^2}\), and \(\sqrt{2}\) stays as \(\sqrt{2}\). This gives us the updated expression: \(\sqrt{7} \cdot \frac{\sqrt{2*3^2}}{\sqrt{2}}\)

Step 2: Simplify the expression further

The second step is to simplify the square roots further by eliminating matching square root in the numerator and denominator. Here, the \(\sqrt{2}\) in \(\sqrt{2*3^2}\) cancels with \(\sqrt{2}\) in the denominator, which simplifies the expression to: \(\sqrt{7} \cdot 3\)

Step 3: Simplify the expression to the simplest form

The final step is to multiply \(\sqrt{7}\) with 3 to get the simplest form of the expression. The final result is: \(3\sqrt{7}\)

Key Concepts

Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime numbers. It's like breaking a number down to its smallest building blocks. Each prime number is a number greater than 1 that cannot be divided by any other numbers except for 1 and itself.
In the context of simplifying radical expressions, prime factorization helps find which numbers can be simplified under the square root.

• For example, consider 18. To find the prime factors, divide by the smallest prime (2): \(18 \div 2 = 9\).
• Then, continue with 9 using the next smallest prime (3): \(9 \div 3 = 3\).
• You get: \(18 = 2 \times 3^2\).

Prime factorization is particularly useful when dealing with square roots, as it allows for easier identification of perfect squares that can be moved outside the radical sign.

Simplifying Square Roots

Simplifying a square root means rewriting it in its simplest form. To do this, we need to identify and extract perfect squares from under the square root. A perfect square is a number that is the square of an integer, like 4 (\(2^2\)) or 9 (\(3^2\)).
For instance, when simplifying \(\sqrt{18}\), using prime factorization, we find \(18 = 2 \times 3^2\).

• The square root of 18 is \(\sqrt{2 \times 3^2}\).
• The \(3^2\) can be extracted as a perfect square so, \(\sqrt{3^2} = 3\).
• Thus, \(\sqrt{18} = 3\sqrt{2}\).

This process helps in reducing complicated-looking square roots into simpler terms, making calculations easier and expressions neater.

Rationalizing Denominators

Rationalizing the denominator involves eliminating any irrational numbers, such as square roots, from the bottom of a fraction. The goal is to make the denominator a rational number.
In the given exercise, rationalizing isn’t needed after simplification because \(\sqrt{2}\) in the denominator cancels with \(\sqrt{2}\) in the numerator of \(\sqrt{18}\).

• After cancellation, the expression becomes \( \sqrt{7} \cdot 3\), simplifying the entire expression further to \(3\sqrt{7}\).

Even though rationalizing was unnecessary here due to cancellation, understanding this concept is important as it comes into play in more complex problems. Sometimes, multiplying both the numerator and the denominator by a suitable expression is needed to clear irrational components in practice.