Problem 300 - Elementary Algebra

Question

In the following exercises, simplify. $$ \sqrt{3} \cdot \sqrt{21} $$

Step-by-Step Solution

Answer

\( \sqrt{3} \cdot \sqrt{21} = 3\sqrt{7} \)

Step 1: Use the Property of Square Roots

Recall the multiplication property of square roots: \ \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). This property allows combining the square roots.

Step 2: Combine the Square Roots

Apply the property to combine the expression: \ \( \sqrt{3} \cdot \sqrt{21} = \sqrt{3 \cdot 21} \).

Step 3: Multiply the Numbers Inside the Square Root

Multiply the numbers inside the square root: \ \( 3 \cdot 21 = 63 \). This gives us \ \( \sqrt{63} \).

Step 4: Simplify the Square Root if Possible

Check if \ \( \sqrt{63} \) can be simplified further. Factor 63 into its prime factors: \ \( 63 = 3^2 \cdot 7 \). Using the property of square roots for multiplication, separate it: \ \( \sqrt{63} = \sqrt{3^2 \cdot 7} = \sqrt{3^2} \cdot \sqrt{7} = 3\sqrt{7} \).

Key Concepts

Multiplication Property of Square Roots

The multiplication property of square roots is a handy tool. This property states that the product of two square roots is the square root of their product.
In other words: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\).
This property allows us to combine square roots into a single square root, which can make simplifying expressions easier.
For example, in the problem, we used this property to combine \(\sqrt{3} \cdot \sqrt{21}\) into \(\sqrt{3 \cdot 21}\).
Remember: This property only works when both expressions are inside square roots.

Prime Factorization

Prime factorization involves breaking down a number into its prime number components. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
For instance, the number 63 can be broken down into its prime factors as follows:
1. Divide 63 by the smallest prime number (which is 3). You get 21.
2. Divide 21 by 3 again to get 7, a prime number.
So, prime factorization of 63 is \(3^2 \cdot 7\).
Using prime factorization helps in simplifying square roots because you can easily see the pairs of prime numbers inside the square root.

Square Root Simplification

Simplifying square roots means reducing them to their simplest form. When a number inside a square root has a pair of identical factors, you can simplify it.
For example, \(\sqrt{63}\) can be broken down using its prime factors: \(\sqrt{3^2 \cdot 7}\).
According to the properties of square roots, this can be simplified to: \(\sqrt{3^2} \cdot \sqrt{7} = 3\sqrt{7}\).
So, \(\sqrt{63}\) simplifies to \(3\sqrt{7}\).
Square root simplification involves:

• Finding the prime factorization of the number inside the square root.
• Identifying pairs of factors.
• Taking pairs out of the square root and leaving unpaired factors inside.

By understanding and using these steps, you can efficiently simplify any square roots you encounter.