Problem 47 - Intermediate Algebra

Question

Multiply. $$ \sqrt{-2} \sqrt{-12} $$

Step-by-Step Solution

Answer

The product is \(2\sqrt{6}\).

Step 1: Identify the Product of Radicals

First, recognize that you are asked to multiply the two square roots. In general, the product of two radicals, \( \sqrt{a} \) and \( \sqrt{b} \), is \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \, b} \). Thus, \( \sqrt{-2} \times \sqrt{-12} = \sqrt{(-2) \times (-12)} \).

Step 2: Simplify the Product inside the Square Root

Calculate the product \((-2) \times (-12)\). Since negative multiplied by negative is positive, \( (-2) \times (-12) = 24 \). So, the product inside the square root becomes \( \sqrt{24} \).

Step 3: Simplify the Square Root

Now simplify \( \sqrt{24} \). Start by factoring the number 24 into its prime factors: \( 24 = 2^3 \times 3 \). This can be rewritten as \( \sqrt{2^3 \times 3} \). To simplify, pair the factors for square root extraction: \( \sqrt{(2^2) \times 6} = \sqrt{4 \times 6} = \sqrt{4} \cdot \sqrt{6} = 2 \cdot \sqrt{6} \).

Key Concepts

Multiplying Radicals

When multiplying radicals, there's a fundamental rule that makes the process easy: to find the product of two square roots, \( \sqrt{a} \) and \( \sqrt{b} \), you multiply the numbers inside the square roots. This concept can be expressed mathematically as \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).
In practice, this means you multiply the numerical values under each radical sign, and then apply a single square root to the product. For example, in our exercise \( \sqrt{-2} \cdot \sqrt{-12} = \sqrt{(-2) \cdot (-12)} = \sqrt{24} \). The multiplication step transforms two separate radical expressions into one, simplifying calculation and evaluation.
It's important to watch for negative numbers and remember that multiplying two negative numbers results in a positive product, as seen here where \( (-2) \cdot (-12) \) becomes \( 24 \). This rule is crucial for simplifying the resulting expression effectively.

Simplifying Square Roots

Simplifying square roots is about expressing a radical in its simplest form. Once you've acquired the product inside the radical, the next step is to simplify, as we did with \( \sqrt{24} \) in the exercise.
To simplify the square root of a number, you look for the largest perfect square factor. The perfect square factor will allow you to break down and simplify the square root.
In the example of \( \sqrt{24} \), we find that \( 24 \) can be expressed in terms of factors. The prime factorization of \( 24 \) is \( 2^3 \times 3 \). Noticing \( 4 \) is a perfect square among these factors, \( 4 = 2^2 \), we simplify:

• Write \( 24 \) as \( 4 \times 6 \).
• So, \( \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \cdot \sqrt{6} = 2 \cdot \sqrt{6} \).

This simplification process involves reducing the square root by extracting the square root of the perfect square (e.g., \( \sqrt{4} = 2 \)), thus simplifying the expression as much as possible.

Prime Factorization

Prime factorization is a method used to express a number as the product of its prime numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself. This method is pivotal when simplifying square roots, as seen in the \( \sqrt{24} \) example.
To perform prime factorization:

• Begin with the smallest prime number (2) and divide the number if possible.
• Continue dividing the quotient by the smallest prime number possible until only prime factors remain.

Using the number \( 24 \), the factorization is:

• Divide \( 24 \) by \( 2 \) to get \( 12 \).
• Divide \( 12 \) by \( 2 \) to get \( 6 \).
• Divide \( 6 \) by \( 2 \) to get \( 3 \), which is prime.

Resulting in \( 24 = 2^3 \times 3 \). This factorization reveals any perfect squares, which are essential for simplifying radicals effectively. By identifying the factor of \( 4 = 2^2 \), you can simplify square roots to their utmost simplicity.