Problem 17 - Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ 2 \sqrt{3} \sqrt{6} $$

Step-by-Step Solution

Answer

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

Step 1: Apply the Property of Square Roots

Use the property of square roots, which states that \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \). In this case, multiply the expressions under the square roots: \( \sqrt{3} \times \sqrt{6} = \sqrt{3 \times 6} \).

Step 2: Simplify the Expression Under the Square Root

Multiply the numbers inside the square root: \( 3 \times 6 = 18 \). So, \( \sqrt{3 \times 6} = \sqrt{18} \).

Step 3: Simplify \( \sqrt{18} \)

Factor \( 18 \) into its prime factors: \( 18 = 2 \times 3^2 \). Use the property \( \sqrt{a^2} = a \) to simplify: \( \sqrt{18} = \sqrt{2 \times 3^2} = 3 \sqrt{2} \).

Step 4: Multiply the Result by 2

Now, multiply the coefficient outside the square root by the result from step 3: \( 2 \times 3 \sqrt{2} = 6 \sqrt{2} \).

Key Concepts

Properties of Square Roots

Square roots have some unique properties that make them fascinating to work with. One fundamental property is that square roots can be multiplied together. For any non-negative real numbers, \( a \) and \( b \), the property \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \) holds true.
This allows us to combine two radical expressions into a single one. Let's break down how this works.

• When you encounter \( \sqrt{3} \times \sqrt{6} \), you can merge them into one radical: \( \sqrt{3 \times 6} \).
• By understanding this property, calculations become simpler, as you're dealing with just one square root instead of two.

Understanding this property lets you simplify expressions efficiently, which is not only useful in exercises but also in real-life applications of algebra.

Prime Factorization

Simplifying square roots often requires us to break numbers down into their prime factors. Prime factorization is the process of expressing a number as the product of its prime numbers.
For example, consider the number 18. Its prime factorization involves breaking it down into 2 and 3, since 18 equals \( 2 \times 3^2 \). This is invaluable when simplifying radicals.

• First, identify all the prime factors. For 18, these are 2 and 3, specifically as \( 2 \times 3^2 \).
• Once primes are identified, use them to simplify the square root further. For instance, \( \sqrt{18} \) equals \( \sqrt{2 \times 3^2} \).
• Apply the property \( \sqrt{a^2} = a \) to simplify further, resulting in \( 3 \sqrt{2} \).

This method of breaking down and simplifying is crucial and makes handling complex square root expressions much easier.

Multiplication of Radicals

Once the properties of square roots and prime factorization are understood, multiplying radicals becomes straightforward. When you multiply radicals, start by combining them under a single square root if possible.
With our expression \( 2 \sqrt{3} \sqrt{6} \):

• First, apply the multiplication property for the radicals: \( \sqrt{3} \times \sqrt{6} = \sqrt{18} \).
• Simplify \( \sqrt{18} \) using prime factorization to get \( 3 \sqrt{2} \).
• Finally, multiply the outside coefficient (2) by the simplified radical: \( 2 \times 3 \sqrt{2} = 6 \sqrt{2} \).

These steps show how multiplication doesn't change the process significantly but rather fits into the system of simplifying radicals. This approach allows for the easy manipulation and simplification of expressions involving square roots.