Problem 279 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

-62 + 55\sqrt{5}

Step 1: Apply the distributive property

Use the distributive property (also known as the FOIL method for binomials) to expand \( (7-2 \sqrt{5})(4+9 \sqrt{5}) \). Multiply each term in the first binomial by each term in the second binomial.

Step 2: Multiply the first terms

Multiply the first terms: \( 7 \times 4 = 28 \).

Step 3: Multiply the outer terms

Multiply the outer terms: \( 7 \times 9\sqrt{5} = 63\sqrt{5} \).

Step 4: Multiply the inner terms

Multiply the inner terms: \( -2\sqrt{5} \times 4 = -8\sqrt{5} \).

Step 5: Multiply the last terms

Multiply the last terms: \( -2\sqrt{5} \times 9\sqrt{5} = -18 \times 5 = -90 \).

Step 6: Combine all the products

Add all the products together: \( 28 + 63\sqrt{5} - 8\sqrt{5} - 90 \).

Step 7: Simplify the expression

Combine like terms: \( 28 - 90 = -62 \) and \( 63\sqrt{5} - 8\sqrt{5} = 55\sqrt{5} \). Thus, the simplified expression is \(-62 + 55\sqrt{5} \).

Key Concepts

Distributive Property

The distributive property helps simplify expressions when you multiply a sum or difference by another sum or difference. It means that each term in one expression is multiplied by each term in the other expression. This is often called the FOIL method when dealing with binomials.

For example, in the expression \( (7 - 2\sqrt{5})(4 + 9\sqrt{5}) \), you distribute each term in the first binomial \( (7 - 2\sqrt{5}) \) to each term in the second binomial \( (4 + 9\sqrt{5}) \).
Here's how it works step by step:

• First: Multiply the first terms of each binomial: \( 7 \times 4 = 28 \).
• Outer: Multiply the outer terms: \( 7 \times 9\sqrt{5} = 63\sqrt{5} \).
• Inner: Multiply the inner terms: \( -2\sqrt{5} \times 4 = -8\sqrt{5} \).
• Last: Multiply the last terms: \( -2\sqrt{5} \times 9\sqrt{5} = -18 \times 5 = -90 \).

These steps cover all combinations of terms, making sure no part of the expression is left out.

Binomial Multiplication

Binomial multiplication is a crucial algebraic operation where two binomials are multiplied together. Binomials are algebraic expressions containing two terms, like \( 7 - 2\sqrt{5} \) or \( 4 + 9\sqrt{5} \). There are specific steps to follow to ensure that every term in the first binomial multiplies with every term in the second binomial.
In our example, to multiply \( (7 - 2\sqrt{5})(4 + 9\sqrt{5}) \), you perform the following:

• Multiply the first terms: \( 7 \times 4 = 28 \).
• Multiply the outer terms: \( 7 \times 9\sqrt{5} = 63\sqrt{5} \).
• Multiply the inner terms: \( -2\sqrt{5} \times 4 = -8\sqrt{5} \).
• Multiply the last terms: \( -2\sqrt{5} \times 9\sqrt{5} = -90 \).

Combine all these products: \( 28 + 63\sqrt{5} - 8\sqrt{5} - 90 \). This method ensures all parts of the binomials are multiplied correctly.

Like Terms

Like terms are terms in an algebraic expression that have identical variable parts and can be combined by addition or subtraction. Only the coefficients (numbers in front of the variables) of like terms are added or subtracted.
In the expression after distributing and multiplying the binomials, you get \[ 28 + 63\sqrt{5} - 8\sqrt{5} - 90 \].
To simplify, combine like terms:

• Constant terms: \( 28 - 90 = -62 \).
• Radical terms: \( 63\sqrt{5} - 8\sqrt{5} = 55\sqrt{5} \).

So, the simplified expression is \ -62 + 55\sqrt{5} \.

Combining like terms makes the expression neater and is essential for simplification in algebra.