Problem 244 - Elementary Algebra

Question

In the following exercises, simplify. $$ (-2 \sqrt{11})(-4 \sqrt{22}) $$

Step-by-Step Solution

Answer

88 \sqrt{2}

Step 1 - Multiply the Constants

First, multiply the constants in the expression. The constants are -2 and -4. $$-2 \times -4 = 8$$

Step 2 - Multiply the Radicals

Next, multiply the radicals together. The radicals are \( \sqrt{11} \) and \( \sqrt{22} \). Using the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \), we get: $$ \sqrt{11} \cdot \sqrt{22} = \sqrt{242} $$

Step 3 - Combine the Results

Now combine the product of the constants and the radicals: $$ 8 \sqrt{242} $$

Step 4 - Simplify the Radical (if possible)

Finally, check if \( \sqrt{242} \) can be simplified further. The prime factors of 242 are 2, 11, and 11. Thus \( 242 = 2 \cdot 11^2 \).Hence, \( \sqrt{242} = \sqrt{2 \cdot 11^2} = 11 \sqrt{2} \). So the simplified form is: $$ 8 \cdot 11 \sqrt{2} = 88 \sqrt{2} $$

Key Concepts

Multiplying Constants

Multiplying constants is an essential step when working with expressions that include numbers and radicals. Constants are the numbers written without a radical sign. For example, in the expression \( -2 \sqrt{11} \), \(-2\) is the constant part. To multiply constants, follow these simple steps:

• Identify the constants in the expression.
• Multiply them directly just as you would with regular numbers.

So for our example, we take \(-2\) and \(-4\). Using multiplication rules:
\(-2 \times -4 = 8\).
Notice that multiplying two negative numbers gives a positive result. This is always the case with negative multiplication.

Multiplying Radicals

After dealing with the constants, we move on to multiplying the radicals. Radicals are the parts of the expression under the root sign, like \( \sqrt{11} \) and \(\b\sqrt{22} \). To multiply radicals:

• Combine the numbers under the radicals using multiplication.
• Use the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).

For example, with \( \sqrt{11} \) and \(\b\sqrt{22} \), we combine them as:
\(\b\sqrt{11} \cdot \sqrt{22} = \sqrt{242} \).
This property allows us to combine the terms easily under a single radical.

Simplifying Square Roots

Simplifying square roots can help us make the final expression much neater. To simplify a square root:

• Factor the number under the radical into its prime factors.
• Look for pairs of prime factors.
• Move pairs of prime factors out from under the radical, as they combine into a single number outside the radical.

In the given example, we need to simplify \( \sqrt{242} \). First, find its prime factors:
\(\b\sqrt{242} = \sqrt{2 \cdot 11^2} \).
Since \( 11^2 \) is a pair, it can be moved outside the radical:
\(\b\sqrt{242} = 11 \sqrt{2} \).
Finally, combine it with the constant we calculated earlier to get the simplified expression: \( 8 \cdot 11 \sqrt{2} = 88 \b\sqrt{2} \).