Problem 275 - Elementary Algebra

Question

In the following exercises, simplify. $$ (9-\sqrt{2})(6+\sqrt{2}) $$

Step-by-Step Solution

Answer

52 + 3\sqrt{2}

Step 1 - Distribute each term in the first binomial across the second binomial using the FOIL method

To simplify \((9-\sqrt{2})(6+\sqrt{2})\), use the FOIL (First, Outer, Inner, Last) method. Multiply the First terms: \(9 \cdot 6 = 54\).

Step 2 - Multiply the Outer terms

Next, multiply the Outer terms: \(9 \cdot \sqrt{2} = 9\sqrt{2}\).

Step 3 - Multiply the Inner terms

Then, multiply the Inner terms: \(-\sqrt{2} \cdot 6 = -6\sqrt{2}\).

Step 4 - Multiply the Last terms

Finally, multiply the Last terms: \(-\sqrt{2} \cdot \sqrt{2} = -2\).

Step 5 - Combine all the products

Now, combine all the products: \(54 + 9\sqrt{2} - 6\sqrt{2} - 2\).

Step 6 - Simplify the expression

Combine like terms to simplify: \(54 - 2 + 9\sqrt{2} - 6\sqrt{2} = 52 + 3\sqrt{2}\).

Key Concepts

FOIL Method

The FOIL method is a technique used to multiply two binomials. FOIL stands for First, Outer, Inner, Last, representing the order in which you multiply the terms.
To understand this method better, let's break it down with the expression \((9-\sqrt{2})(6+\sqrt{2})\).

First, we multiply the First terms: \(9 \times 6 = 54\).

Next, we multiply the Outer terms: \(9 \times \sqrt{2} = 9\sqrt{2}\).

Then, we move to the Inner terms: \(-\sqrt{2} \times 6 = -6\sqrt{2}\).

Finally, we handle the Last terms: \(-\sqrt{2} \times \sqrt{2} = -2\).

These steps help ensure every term from each binomial is multiplied correctly.

Simplifying Expressions

Simplifying expressions involves combining like terms and reducing the expression to its simplest form.
After using the FOIL method on \((9-\sqrt{2})(6+\sqrt{2})\), you get: \[54 + 9\sqrt{2} - 6\sqrt{2} - 2\].

To simplify, first combine the constant terms: \(54 - 2 = 52\).
Next, combine the radical terms: \(9\sqrt{2} - 6\sqrt{2} = 3\sqrt{2}\).

The simplified expression is thus: \(52 + 3\sqrt{2}\).
Always look for like terms to combine, whether they are constants or include radicals.

Radicals in Algebra

When dealing with radicals in algebra, it's important to understand how to manipulate and combine them.

Radicals, often represented by the square root symbol \(\sqrt{}\), can be simplified by finding common terms.
In our example, the terms \(9\sqrt{2}\) and \(-6\sqrt{2}\) are like terms because they share the same radical part: \(\sqrt{2}\).
Combining like radical terms, we sum or subtract their coefficients, resulting in: \(9\sqrt{2} - 6\sqrt{2} = 3\sqrt{2}\).

Remember that when multiplying radicals, if the radicals are the same, the radicands (the values inside the square roots) are multiplied directly: \(-\sqrt{2} \times \sqrt{2} = -2\).