Problem 36 - Intermediate Algebra

Question

Simplify. See Examples 3 and 4 $$ 3 \sqrt{8} $$

Step-by-Step Solution

Answer

The simplified form is \(6 \sqrt{2}\).

Step 1: Prime Factorize Inside the Square Root

Identify and factorize the number inside the square root. For the number 8, its prime factorization is: \(8 = 2 \times 2 \times 2 = 2^3\).

Step 2: Simplify Using Square Root Properties

Use the property \( \sqrt{a^2 \times b} = a \sqrt{b} \) to simplify. With \( 8 = 2^3 \), we have 3 twos; factor out pairs from under the square root: \( \sqrt{8} = \sqrt{2^2 \times 2} = 2 \sqrt{2} \).

Step 3: Multiply by the Coefficient

Multiply the simplified square root expression by the coefficient from the original expression. This results in: \( 3 \times 2 \sqrt{2} = 6 \sqrt{2} \).

Key Concepts

Prime Factorization

When simplifying radical expressions, it's important to understand prime factorization. This process involves breaking down a number into its prime numbers, which are numbers greater than 1 that have no divisors other than 1 and itself. For example, to find the prime factors of 8, you divide it into the smallest prime numbers:

• Start with 8, and since 2 is the smallest prime number, divide 8 by 2 to get 4.
• Next, divide 4 by 2, which results in 2.
• Finally, divide 2 by 2 to get 1.

Thus, the prime factorization of 8 is 2 multiplied by itself three times, or \(2 \times 2 \times 2 = 2^3\). Knowing this allows you to simplify expressions under a square root more easily.

Properties of Square Roots

After prime factorizing the number inside a square root, the properties of square roots can help simplify the expression. A key property is \( \sqrt{a^2 \times b} = a \sqrt{b} \).This property allows you to "take out" any numbers in pairs from underneath the square root. For instance, in the expression \( \sqrt{8} \), with 8 written as \(2^3\), you have:

• Pair two of the threes: \(\sqrt{2^2 \times 2}\).
• Bring out one 2 from the square root: \(2 \sqrt{2}\).

Pairs of identical factors can be extracted from the root, thus simplifying the radical. This fundamental step clears out complexities and brings the expression to its simplest form.

Coefficients in Algebraic Expressions

In algebra, coefficients are the numbers multiplied by variables or radical expressions. In our example, 3 is the coefficient outside \(\sqrt{8}\). Once you simplify the radical part (\(\sqrt{8}\) simplifies to \(2 \sqrt{2}\)), it's time to amplify it by multiplying back in the coefficient:

• Multiply the extracted number (2) from the radical with the original coefficient (3).
• This results in \(3 \times 2 = 6\).

So, when coefficients interact with radicals, simplify the radical first, then incorporate the coefficient to find the final simplified expression. This way, you maintain clarity and precision in algebraic operations.