Problem 282 - Elementary Algebra

Question

In the following exercises, simplify. $$ (2 \sqrt{7}-5 \sqrt{11})(4 \sqrt{7}+9 \sqrt{11}) $$

Step-by-Step Solution

Answer

\[\begin{equation} -439 - 2 \sqrt{77}. \end{equation}\]

Step 1 - Apply the distributive property

Use the distributive property to expand the expression \[\begin{equation} (2 \sqrt{7} - 5 \sqrt{11})(4 \sqrt{7} + 9 \sqrt{11}) \end{equation}\] which is of the form \[\begin{equation} (a - b)(c + d) = ac + ad - bc - bd \end{equation}\].

Step 2 - Multiply the first terms

First, multiply \[\begin{equation} 2 \sqrt{7} \times 4 \sqrt{7} = 8 \times 7 = 56. \end{equation}\]

Step 3 - Multiply the outer terms

Now multiply the outer terms, \[\begin{equation} 2 \sqrt{7} \times 9 \sqrt{11}. \end{equation}\] This equals \[\begin{equation} 18 \sqrt{77}. \end{equation}\]

Step 4 - Multiply the inner terms

Next, multiply the inner terms, \[\begin{equation} -5 \sqrt{11} \times 4 \sqrt{7}. \end{equation}\] This equals \[\begin{equation} -20 \sqrt{77}. \end{equation}\]

Step 5 - Multiply the last terms

Lastly, multiply the last terms, \[\begin{equation} -5 \sqrt{11} \times 9 \sqrt{11} = -45 \times 11. \end{equation}\] This simplifies to \[\begin{equation} -495. \end{equation}\]

Step 6 - Combine like terms

Combine all the terms found in the previous steps: \[\begin{equation} 56 + 18 \sqrt{77} - 20 \sqrt{77} - 495. \end{equation}\] Combine the \sqrt{77} terms: \[\begin{equation} 56 - 495 - 2 \sqrt{77}. \end{equation}\]

Step 7 - Simplify

Simplify the constants first: \[\begin{equation} 56 - 495 = -439. \end{equation}\] The final simplified expression is thus \[\begin{equation} -439 - 2 \sqrt{77}. \end{equation}\]

Key Concepts

Distributive Property

To simplify the expression \( (2 \sqrt{7} - 5 \sqrt{11})(4 \sqrt{7} + 9 \sqrt{11}) \), we first apply the distributive property. This property allows us to expand the expression by multiplying each term in the first parenthesis by each term in the second parenthesis. The general form we use is: \( (a - b)(c + d) = ac + ad - bc - bd \)

By distributing each term, we ensure that we do not miss any multiplication components, which will later help us combine and simplify the terms properly.

Multiplying Radicals

When multiplying radicals like \(2 \sqrt{7} \times 4 \sqrt{7} \), it's important to multiply both the coefficients (numbers outside the roots) and the radicands (numbers inside the roots) separately. This means:

• First, multiply the coefficients: 2 and 4, resulting in 8.
• Then, multiply the radicands: \(\sqrt{7} \times \sqrt{7} = \sqrt{49} = 7 \).

This gives us: \( 2 \sqrt{7} \times 4 \sqrt{7} = 8 \times 7 = 56 \)

Using a similar approach, \( 2 \sqrt{7} \times 9 \sqrt{11} = 18 \sqrt{77} \), and \(-5 \sqrt{11} \times 9 \sqrt{11} = -45 \times 11 = -495 \). Understanding how to multiply these radicals will help simplify the rest of the expression.

Combining Like Terms

Once you have expanded the expression by multiplying each term, some of the terms will have similar components and can be combined. For instance, in the expanded expression:

• \( 56 + 18 \sqrt{77} - 20 \sqrt{77} - 495 \)

The terms with \(\sqrt{77}\) can be combined:

• \(18 \sqrt{77} - 20 \sqrt{77} = (18 - 20) \sqrt{77} = -2 \sqrt{77} \).

Combining like terms simplifies the expression further, as we get: \ 56 - 495 - 2 \sqrt{77} \.

Simplifying Algebraic Expressions

After combining like terms, the final step is to simplify the constants and put together the expression in its simplest form. For the given problem, we arrive at:

• \( 56 - 495 - 2 \sqrt{77} \)
• Simplify the constants: \( 56 - 495 = -439 \)

Therefore, the simplified expression is: \ -439 - 2 \sqrt{77} \.

Simplifying algebraic expressions involves applying these steps effectively while ensuring that all like terms are combined and constants are simplified properly.