Problem 38 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \((5 \sqrt{2}-4 \sqrt{6})(2 \sqrt{8}+\sqrt{6})\)

Step-by-Step Solution

Answer

The simplified form is \(16 - 22\sqrt{3}\).

Step 1: Rewrite the Radicals

First, we rewrite the radicals so that each has only prime factored radicands: \(2\sqrt{8} = 2\sqrt{4 \times 2} = 2 \cdot 2 \sqrt{2} = 4\sqrt{2}\). Hence, the expression becomes \((5\sqrt{2} - 4\sqrt{6})(4\sqrt{2} + \sqrt{6})\).

Step 2: Apply the Distributive Property

Apply the distributive property (FOIL method) to expand the product: \((5\sqrt{2})(4\sqrt{2}) + (5\sqrt{2})(\sqrt{6}) - (4\sqrt{6})(4\sqrt{2}) - (4\sqrt{6})(\sqrt{6})\).

Step 3: Simplify Each Term

Simplify each term:1. \( (5\sqrt{2})(4\sqrt{2}) = 20(\sqrt{4}) = 20 \times 2 = 40\).2. \( (5\sqrt{2})(\sqrt{6}) = 5\sqrt{12} = 5\sqrt{4\times3} = 5 \times 2\sqrt{3} = 10\sqrt{3}\).3. \( -(4\sqrt{6})(4\sqrt{2}) = -16\sqrt{12} = -16 \times 2\sqrt{3} = -32\sqrt{3}\).4. \( -(4\sqrt{6})(\sqrt{6}) = -4\sqrt{36} = -4 \times 6 = -24\).

Step 4: Combine Like Terms

Combine like terms to simplify the expression: \(40 - 24 + 10\sqrt{3} - 32\sqrt{3}\).This simplifies to \(16 - 22\sqrt{3}\).

Key Concepts

Distributive Property

The distributive property is quite useful when dealing with algebraic expressions, especially when simplifying radicals. It allows us to multiply a single term by two or more terms inside a parenthesis. In this exercise, we applied the distributive property using the FOIL method, which is perfect for expanding the product of two binomials. FOIL stands for First, Outer, Inner, and Last, referencing the pairs of terms you multiply together from each set of parentheses.

For our example,

• First: Multiply the first terms from each binomial. i.e., \[ (5\sqrt{2} \times 4\sqrt{2}) \]
• Outer: Multiply the outer terms. i.e., \[ (5\sqrt{2} \times \sqrt{6}) \]
• Inner: Multiply the inner terms. i.e., \[ (-4\sqrt{6} \times 4\sqrt{2}) \]
• Last: Multiply the last terms from each binomial. i.e., \[ (-4\sqrt{6} \times \sqrt{6}) \]

By expanding the expression using FOIL, each pair is multiplied to give new terms, which can often be further simplified.

Prime Factoring

Prime factoring is the process of breaking down numbers into their prime components, which is very useful in simplifying radicals. When simplifying expressions under a square root, identifying if part of the radicand (the number inside the radical sign) can be broken down into perfect squares can greatly simplify things.

For example, let's consider the term \( \sqrt{8} \):

• The number 8 can be expressed as \( 4 \times 2 \).
• Since 4 is \( 2^2 \) (a perfect square), it comes out of the root as 2.
• This simplifies \( 2\sqrt{8} \) to \( 4\sqrt{2} \).

Prime factorization identifies such simplifications, making expressions tidier and easier to work with for further calculations. Applying prime factoring helps us rewrite radicals in a way that reveals components which are perfect squares.

Combining Like Terms

Combining like terms is an essential algebraic process that helps simplify expressions, particularly those involving addition or subtraction. Like terms in algebra are terms which have identical variable components raised to the same powers. In our main exercise, after multiplying out the expression, some terms can be combined.

For instance:

• \( 10\sqrt{3} \)
• and \(-32\sqrt{3} \)

These are like terms because both contain the \( \sqrt{3} \) component. Therefore, we subtract the coefficients (\( 10 \) and \( -32 \)) to get \(-22\sqrt{3} \).

Carefully identifying and combining like terms leads to simplified expressions, making them neater and letting us easily interpret or utilize them for further mathematical operations.

Algebraic Expressions

Algebraic expressions consist of constants, variables, and operations. Understanding and manipulating these expressions is crucial to solve many mathematics problems, including radical simplification.

An expression like \( (5\sqrt{2} - 4\sqrt{6})(4\sqrt{2} + \sqrt{6}) \) involves multiple elements:

• Each term within the parentheses is part of a radical expression.
• The operations between these terms are addition, subtraction, and multiplication.

Mastering algebraic manipulation, such as using the distributive property and combining like terms, transforms algebraic expressions into their simplest form. This process not only makes expressions simpler but also aids in solving equations and inequalities involving similar components in further mathematical pursuits.