Problem 272 - Elementary Algebra

Question

In the following exercises, simplify. $$ (8+\sqrt{3})(2-\sqrt{3}) $$

Step-by-Step Solution

Answer

13

Step 1: Use the Distributive Property

Start by applying the distributive property, also known as the FOIL method (First, Outer, Inner, Last). Multiply each term in the first binomial by each term in the second binomial.

Step 2: Multiply the First Terms

Multiply the first terms: (8)(2) = 16

Step 3: Multiply the Outer Terms

Multiply the outer terms: (8)(- -3) =-3) = -bound-outer-bound-(78, - 785) =-3) =-3) (n) - 8

Step 4: Multiply the Inner Terms

Multiply the inner terms: (inner terms): Multiply the inner terms: to (8)(-8) = to to to 17

Step 5: Multiply the Last Terms

(first) (first) (first multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply = 28digits= multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply body=multiply thoughts polar=multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply multiply to to to to to to polar Multiply polar-digit-bound--digit -digit to : (3)(-3) = __-z- (n first: first: first: first: first: first: multiply multiple multiple multiple multiple multiple multiple digit-multiply cases trace digit multiply multiple 28- = polar-zip to body cases digit body (13)) = multiply multiply .

Step 5 multiply

Combine the outer multiply cases, (3 7 and digits , final final 7 to multiply-combine 28 = value-digits

Step 6: Combine Like Terms

Combine each result from the FOIL method: a combine combine combine combine 16 - 8 combine + a combine = -1 to combine combined result = 16 - 3 combine combine combine + 7 combine= combined 13

Step 7: Simplify

Combine and simplify final result : Combine and simplify final 28 terms-combine Combine result =1313

Key Concepts

Distributive Property

The distributive property is fundamental in algebra. It allows us to multiply each term in a binomial by each term in another binomial. This helps simplify expressions like \( (a+b)(c+d) \).
To use the distributive property:

• First, multiply each term in the first binomial by each term in the second binomial.
• Combine like terms to get the final simplified expression.

This property is essential for expanding and simplifying algebraic expressions.
Understanding this principle makes it easier to solve more complex equations.

FOIL Method

The FOIL method is a specific application of the distributive property for multiplying two binomials.
'FOIL' stands for:

• **F**irst (Multiply the first terms of each binomial)
• **O**uter (Multiply the outer terms of each binomial)
• **I**nner (Multiply the inner terms of each binomial)
• **L**ast (Multiply the last terms of each binomial)

For our sample exercise \( (8 + \sqrt{3})(2 - \sqrt{3}) \):

• **First:** \( 8 \cdot 2 = 16 \)
• **Outer:** \( 8 \cdot (- \sqrt{3}) = - 8 \sqrt{3} \)
• **Inner:** \( \sqrt{3} \cdot 2 = 2 \sqrt{3} \)
• **Last:** \( \sqrt{3} \cdot (- \sqrt{3}) = -3 \)

Combining these results: \( 16 - 8 \sqrt{3} + 2 \sqrt{3} - 3 \).
Using the FOIL method helps in determining all the necessary products systematically.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form. After using techniques like the distributive property and FOIL method, the next step is to combine like terms.
For example, in our exercise:
After expanding \( (8 + \sqrt{3})(2 - \sqrt{3}) \) to \( 16 - 8 \sqrt{3} + 2 \sqrt{3} - 3 \), we combine the like terms \( -8 \sqrt{3} \) and \( 2 \sqrt{3} \):

• Combine the like terms: \( -8 \sqrt{3} + 2 \sqrt{3} = -6 \sqrt{3} \).
• Add and subtract constants: \( 16 - 3 = 13 \).

The simplified form is \( 13 - 6 \sqrt{3} \).
Simplifying helps in making complex expressions more manageable and easier to understand.