Problem 42 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \((\sqrt{8}-3 \sqrt{10})(2 \sqrt{8}-6 \sqrt{10})\)

Step-by-Step Solution

Answer

The simplest radical form is \(196 - 48\sqrt{5}\).

Step 1: Expand the Expression Using the Distributive Property

Apply the distributive property (FOIL method) to expand the expression. Multiply each term in the first binomial by each term in the second binomial. \((\sqrt{8} - 3\sqrt{10})(2\sqrt{8} - 6\sqrt{10}) = \sqrt{8} \cdot 2\sqrt{8} + \sqrt{8} \cdot (-6\sqrt{10}) - 3\sqrt{10} \cdot 2\sqrt{8} - 3\sqrt{10} \cdot (-6\sqrt{10})\)

Step 2: Simplify Each Term

Simplify each term by multiplication and simplifying radicals:1. \(\sqrt{8} \cdot 2\sqrt{8} = 2(\sqrt{8} \cdot \sqrt{8}) = 2\cdot 8 = 16\)2. \(\sqrt{8} \cdot (-6\sqrt{10}) = -6(\sqrt{8} \cdot \sqrt{10}) = -6\sqrt{80}\)3. \(-3\sqrt{10} \cdot 2\sqrt{8} = -6(\sqrt{10} \cdot \sqrt{8}) = -6\sqrt{80}\)4. \(-3\sqrt{10} \cdot (-6\sqrt{10}) = 18(\sqrt{10} \cdot \sqrt{10}) = 18\times 10 = 180\)

Step 3: Combine Like Terms

Add and combine the like terms in the expression. The expression becomes: \[16 - 6\sqrt{80} - 6\sqrt{80} + 180\]Combine the constant terms and the like radical terms:\[16 + 180 - 12\sqrt{80}\]

Step 4: Simplify the Radical

Simplify the radical \(\sqrt{80}\)\[\sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}\]Substitute back into the expression:\[16 + 180 - 12\times 4\sqrt{5}\]\[16 + 180 - 48\sqrt{5}\]

Step 5: Finalize the Simplified Expression

Combine the constant terms to obtain:\[196 - 48\sqrt{5}\]This is the final simplified expression.

Key Concepts

Distributive Property

The Distributive Property is a fundamental concept in algebra. It allows us to multiply a single term by each term within a parenthesis. When you see an expression like \((a+b)(c+d)\), the Distributive Property helps by breaking it down into simpler parts like \(a \cdot c + a \cdot d + b \cdot c + b \cdot d\). This property is essential when dealing with multiple terms in an expression.

• It simplifies calculations.
• It ensures each term is accounted for during multiplication.
• It helps maintain the accuracy of algebraic operations.

Understanding the Distributive Property makes it easier to tackle more complex expressions and ensures precise arithmetic operations.

FOIL Method

The FOIL Method is a technique used to simplify the multiplication of two binomials. The term "FOIL" stands for First, Outer, Inner, and Last, which refers to the order in which you multiply the terms. Let's take the expression \((a+b)(c+d)\) again:

• First: Multiply the first terms in each binomial, \(a \cdot c\).
• Outer: Multiply the outer terms, \(a \cdot d\).
• Inner: Multiply the inner terms, \(b \cdot c\).
• Last: Multiply the last terms in each binomial, \(b \cdot d\).

Adding all these products together gives us the fully expanded form.
The FOIL Method is a quick and systematic way to ensure that all terms are properly distributed and simplifies the process of opening up binomials.

Simplifying Radicals

Simplifying Radicals involves reducing a square root or other radical expression to its simplest form. Sometimes, numbers under a radical sign can be broken into smaller factors, making them easier to work with. For example:
To simplify \(\sqrt{80}\), notice that \(80 = 16 \times 5\). We can rewrite this as \(\sqrt{16 \times 5} = \sqrt{16} \cdot \sqrt{5}\). Since \(\sqrt{16} = 4\), the simplified form is \(4\sqrt{5}\).
Radicals are fully simplified when:

• There are no perfect squares under the radical (other than 1).
• The expression does not contain a fraction under the radical sign.
• The radical is not in a denominator.

Simplifying radicals makes it much easier to work with algebraic expressions and can significantly reduce the complexity of mathematics problems.

Combining Like Terms

Combining Like Terms is the process of simplifying expressions by merging terms that have identical variable parts. In algebra, like terms are terms that have the same variable raised to the same power. Take the expression:
\[16 - 6\sqrt{80} - 6\sqrt{80} + 180\].
The terms \(16\) and \(180\) are constant terms and combine directly to give \(196\). Similarly, the terms \(-6\sqrt{80}\) are alike, and when combined, give \(-12\sqrt{80}\). It's important to simplify radicals first, if possible, before combining.

• Look for terms with identical variables and powers.
• Add or subtract the coefficients while keeping the variable part unchanged.
• This process reduces expressions into simpler, more manageable forms.

Combining like terms streamlines expressions and is a foundational skill in algebra for solving equations efficiently.