Question

Simplify each expression. Assume that all variables are positive when they appear. $$(3 \sqrt{6})(2 \sqrt{2})$$

Short answer

12 \text{sqrt}(3)

Step-by-step solution

- Multiply the coefficients

Multiply the numerical coefficients outside the square roots. In this case, multiply 3 and 2:\(3 \times 2 = 6\)

- Multiply the radicands

Multiply the numbers inside the square roots (the radicands) together. In this case, multiply 6 and 2:\(\text{inside the square root:} \, 6 \times 2 = 12\)

- Combine the results

Combine the results from steps 1 and 2 to form a single expression:\(6 \times \text{sqrt}(12) = 6 \text{sqrt}(12)\)

- Simplify the radicand

Break down the number inside the square root (12) into its prime factors to simplify further:\(12 = 4 \times 3 = 2^2 \times 3\)So, \(\text{sqrt}(12) = \text{sqrt}(4 \times 3)\)Which simplifies to \(\text{sqrt}(4) \times \text{sqrt}(3) = 2 \text{sqrt}(3)\)

- Multiply the simplified radicand

Now, multiply the result from step 4 by the coefficient from step 3:\(6 \times 2 \text{sqrt}(3) = 12 \text{sqrt}(3)\)

Key concepts

Multiplying Coefficients

When simplifying radical expressions, the first step is to multiply the coefficients, which are the numbers outside the square root symbols. In our given problem, we have the coefficients 3 and 2. To multiply these coefficients, simply perform the multiplication:
\( 3 \times 2 = 6 ... \)
By multiplying the coefficients first, we simplify the expression one step at a time and keep our calculations organized.

Multiplying Radicands

After multiplying the coefficients, we move on to the radicands. The radicand is the number or expression inside the square root. In our problem, we have square roots of 6 and 2. To handle this, multiply the radicands together:
\( \text{inside the square root:} \, 6 \times 2 = 12 ... \)
This step helps us combine the numbers under one single square root, making it easier to simplify later.

Simplifying Square Roots

Now that we have combined our radicands, it's time to simplify the square root. The expression we have is \( 6 \text{sqrt}(12) \) ... To simplify the square root of 12, we need to break it down into prime factors. Prime factorization of 12 is:
\( 12 = 4 \times 3 = 2^2 \times 3 \)
This allows us to write the square root of 12 as a product of simpler square roots:
\[ \text{sqrt}(12) = \text{sqrt}(4 \times 3) \]
Simplifying further:
\[ \text{sqrt}(4) \times \text{sqrt}(3) = 2 \text{sqrt}(3) \]
So the expression 6 \text{sqrt}(12) becomes:
\[ 6 \times 2 \text{sqrt}(3) \]

Prime Factorization

Prime factorization is a vital tool for simplifying square roots. It involves breaking down a composite number into a product of prime numbers. For example, with the number 12, we find its prime factors by seeing which numbers multiply together to give 12.
The prime factors of 12 are:
\[ 12 = 2 \times 2 \times 3 \]
or
\[ 12 = 2^2 \times 3 \]
Recognizing the square of 2 (\(2^2\)) helps us simplify further, since \( \text{sqrt}(2^2) = 2 \) ... This method makes it easier to turn the radicand into a simpler format.