Problem 285 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

18 + 7\sqrt{w} - w.

Step 1: Identify the Binomials

Recognize that the expression \( (9-\sqrt{w})(2+\sqrt{w}) \) is a product of two binomials.

Step 2: Apply the Distributive Property

Use the distributive property (also known as FOIL for binomials): \( (a+b)(c+d)=ac+ad+bc+bd \)

Step 3: Multiply the First Terms

Multiply the first terms of each binomial: \( 9 \times 2 = 18 \)

Step 4: Multiply the Outer Terms

Multiply the outer terms of each binomial: \( 9 \times \sqrt{w} = 9\sqrt{w} \)

Step 5: Multiply the Inner Terms

Multiply the inner terms of each binomial: \( -\sqrt{w} \times 2 = -2\sqrt{w} \)

Step 6: Multiply the Last Terms

Multiply the last terms of each binomial: \( -\sqrt{w} \times \sqrt{w} = -w \)

Step 7: Combine Like Terms

Combine the results from all multiplications: \( 18 + 9\sqrt{w} - 2\sqrt{w} - w = 18 + 7\sqrt{w} - w \)

Key Concepts

Distributive Property

The Distributive Property is a fundamental algebraic principle that allows us to distribute multiplication over addition or subtraction. It ensures that each term inside a set of parentheses is multiplied by a term outside of it. This property is also essential when working with binomials.
For example, if you have the term \(a(b + c)\), using the Distributive Property you would multiply both \(b\) and \(c\) by \(a\) to get \(ab + ac\).
This principle is often used in more complex expressions to simplify calculations and can also be called the FOIL Method when applied specifically to binomials.

FOIL Method

The FOIL Method is a special case of the Distributive Property, used specifically for multiplying two binomials. The acronym FOIL stands for First, Outer, Inner, Last, which are the terms that need to be multiplied together.
Let's break it down using our example \( (9 - \sqrt{w})(2 + \sqrt{w}) \).

• First: Multiply the first terms of each binomial: \( 9 \times 2 = 18 \).
• Outer: Multiply the outer terms: \( 9 \times \sqrt{w} = 9\sqrt{w} \).
• Inner: Multiply the inner terms: \( -\sqrt{w} \times 2 = -2\sqrt{w} \).
• Last: Multiply the last terms: \( -\sqrt{w} \times \sqrt{w} = -w \).

By applying these steps systematically, you can break down complex binomials into simpler terms, making them much easier to combine and simplify.

Combining Like Terms

Combining Like Terms is a method in algebra used to simplify expressions, where you only add or subtract terms that have the same variable and the same exponent.
After using the FOIL Method, you typically get several terms that need to be simplified. Let's continue with our previous example \( 18 + 9 \sqrt{w} - 2 \sqrt{w} - w \). We need to combine all like terms.

• Identify the like terms: \( +9 \sqrt{w} \) and \( -2 \sqrt{w} \), and combine them.
• \( +9 \sqrt{w} - 2 \sqrt{w} = 7 \sqrt{w} \).

Now, we can rewrite the expression as \( 18 + 7 \sqrt{w} - w \).
This method ensures that the final expression is as simple as possible, making it easier to interpret and use in further calculations.