Problem 32 - Elementary Algebra

Question

For the following problems, simplify the expressions. $$ (6 r+\sqrt{2 s})(4 r+\sqrt{2 s}) $$

Step-by-Step Solution

Answer

Answer: The simplified expression is \(24r^2 + 10r\sqrt{2s} + 2s\).

Step 1: Identify the binomials

We have the expression \((6r + \sqrt{2s})(4r + \sqrt{2s})\). Here, the two binomials to be multiplied are \((6r+\sqrt{2s})\) and \((4r+\sqrt{2s})\).

Step 2: Apply the FOIL method

To multiply these two binomials, we will use the FOIL method. FOIL stands for First, Outer, Inner, and Last. This means we will multiply the First terms, then the Outer terms, then the Inner terms, and finally the Last terms of the binomials.

Step 3: Multiply the First terms

First, we will multiply the first terms of the binomials: \((6r)(4r) = 24r^2\).

Step 4: Multiply the Outer terms

Next, we will multiply the outer terms of the binomials: \((6r)(\sqrt{2s}) = 6r\sqrt{2s}\).

Step 5: Multiply the Inner terms

Now, we will multiply the inner terms of the binomials: \((\sqrt{2s})(4r) = 4r\sqrt{2s}\).

Step 6: Multiply the Last terms

Finally, we will multiply the last terms of the binomials: \((\sqrt{2s})(\sqrt{2s}) = 2s\).

Step 7: Combine the terms

Now we need to combine all the terms that we got from Steps 3 to 6: \(24r^2 + 6r\sqrt{2s} + 4r\sqrt{2s} + 2s\).

Step 8: Simplify the expression

We can combine the terms with the same variable and root (6r\sqrt{2s} and 4r\sqrt{2s}) to simplify the expression: \(24r^2 + (6r+4r)\sqrt{2s} + 2s\). This simplifies to \(24r^2 + 10r\sqrt{2s} + 2s\). The simplified expression is: $$24r^2 + 10r\sqrt{2s} + 2s.$$

Key Concepts

Binomial Multiplication

Binomial multiplication involves the process of multiplying two expressions, each containing two terms. In this context, think of binomials as mathematical phrases composed of terms separated by a plus or minus sign. The task is not simply to distribute terms, but to consider the interactions between each part of the binomials.

It's crucial to understand that when you multiply one binomial by another, you ensure every term in the first binomial multiplies each term in the second binomial. This systematic multiplication process guarantees that no interactions are overlooked. In our exercise, the two binomials are

• (6r + \(\sqrt{2s}\))
• (4r + \(\sqrt{2s}\))

By comprehending binomial multiplication, we can accurately combine these terms and progress toward a simplified expression.

FOIL Method

The FOIL method is a systematic way to multiply two binomials. The acronym stands for First, Outer, Inner, and Last. The strategy focuses on the order of multiplication to ensure all necessary products are calculated.

- **First:** Multiply the first terms of each binomial.- **Outer:** Multiply the outermost terms.- **Inner:** Multiply the innermost terms.- **Last:** Multiply the last terms of both binomials.
In our problem, using FOIL on the binomials

• (6r + \(\sqrt{2s}\)) and
• (4r + \(\sqrt{2s}\))

leads us through a structured multiplication process:

• First: \((6r)(4r) = 24r^2\)
• Outer: \( (6r)(\sqrt{2s}) = 6r\sqrt{2s} \)
• Inner: \((\sqrt{2s})(4r) = 4r\sqrt{2s}\)
• Last: \((\sqrt{2s})(\sqrt{2s}) = 2s\)

By calculating each step, the FOIL method helps in organizing your work and ensures all interactions between terms are captured correctly.

Simplifying Expressions

Simplifying expressions involves combining like terms and making sure the expression is as concise as possible. After multiplying our binomials using the FOIL method, we're left with the expression \[24r^2 + 6r\sqrt{2s} + 4r\sqrt{2s} + 2s\]

To simplify, focus on terms that can be combined. The terms \(6r\sqrt{2s}\) and \(4r\sqrt{2s}\) have similar parts: both share the variable \(r\sqrt{2s}\). Combining them results in:\[(6r\sqrt{2s} + 4r\sqrt{2s}) = 10r\sqrt{2s}\]Now, the entire expression simplifies to:\[24r^2 + 10r\sqrt{2s} + 2s\]Breaking down and methodically organizing like terms in this manner aids in reaching the simplest form of any given expression.