Problem 64 - A Graphical Approach to College Algebra

Question

Simplify each expression, assuming that all variables represent nonnegative real numbers. $$9 \sqrt{8 k}+3 \sqrt{18 k}-\sqrt{32 k}$$

Step-by-Step Solution

Answer

The simplified expression is \(23 \sqrt{2k}\).

Step 1: Identify Like Terms

First, we will identify the terms that can be combined by looking for expressions under the radical that can be simplified similarly. The expression is:\[9 \sqrt{8k} + 3 \sqrt{18k} - \sqrt{32k}\]

Step 2: Simplify Individual Radicals

Simplify each radical individually:- \(\sqrt{8k} = \sqrt{4 \times 2k} = \sqrt{4} \cdot \sqrt{2k} = 2\sqrt{2k}\)- \(\sqrt{18k} = \sqrt{9 \times 2k} = \sqrt{9} \cdot \sqrt{2k} = 3\sqrt{2k}\)- \(\sqrt{32k} = \sqrt{16 \times 2k} = \sqrt{16} \cdot \sqrt{2k} = 4\sqrt{2k}\)

Step 3: Substitute Simplified Radicals into Expression

Replace the radicals in the original expression with their simplified forms:\[9(2 \sqrt{2k}) + 3(3 \sqrt{2k}) - 4 \sqrt{2k}\]

Step 4: Combine Like Terms

Combine the terms by factoring out the common \(\sqrt{2k}\):\[(9 \times 2 + 3 \times 3 - 4) \sqrt{2k}\]Calculate the coefficients:\[18 + 9 - 4 = 23\]

Step 5: Final Simplified Expression

The simplified expression is:\[23 \sqrt{2k}\]

Key Concepts

Combining Like Terms

When you're working on simplifying expressions, a crucial step is combining like terms. Like terms in algebraic expressions are terms that have the same variable part. This means they must have identical variable factors, including their exponents.
For example, in the expression \(9 \sqrt{8k} + 3 \sqrt{18k} - \sqrt{32k}\), the terms \(\sqrt{8k}\), \(\sqrt{18k}\), and \(\sqrt{32k}\) are like terms because they all contain the radical \(\sqrt{k}\). The variable part under the square roots is the same, which allows them to be combined after the radicals are simplified to have the same term.
To combine these like terms effectively:

• Ensure the variable and radical parts are identical after simplification.
• Combine their coefficients as you would with regular numbers or like variables.

After simplifying individual radicals, treat the term \(\sqrt{2k}\) as a common factor and focus on the numerical coefficients. This consolidation helps greatly in reducing and simplifying expressions to form a cleaner mathematical statement.

Simplifying Radicals

Simplifying radicals is a process of rewriting a radical expression in its simplest form. This involves reducing the expression so that any perfect square factors are removed from under the radical. Here’s how you simplify radicals such as \(\sqrt{8k}\), \(\sqrt{18k}\), and \(\sqrt{32k}\):
First, you’ll want to find factors of the radicand that are perfect squares. For example, with \(\sqrt{8k}\), recognize that \(8 = 4 \times 2\) and \(4\) is a perfect square, allowing us to simplify to \(2\sqrt{2k}\).
Similarly,

• For \(\sqrt{18k}\), break it into \(9 \times 2k\) where 9 is a perfect square, simplifying it to \(3\sqrt{2k}\).
• For \(\sqrt{32k}\), use \(16 \times 2k\) since 16 is a perfect square, simplifying to \(4\sqrt{2k}\).

You achieve this by applying the property \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\). By simplifying individual radicals, you make combining them much easier. This simplification is essential for accurately solving problems and reducing expression complexity.

Algebraic Expressions

Algebraic expressions form the foundation of algebra and include numbers, variables, and arithmetic operations. Expressions like the one in this exercise, \(9 \sqrt{8k} + 3 \sqrt{18k} - \sqrt{32k}\), combine these components. Understanding how to manipulate algebraic expressions is key to solving them.
Here are fundamentals to remember:

• Identify parts of the expression: constants, coefficients, radicals, and variables.
• Understand the order of operations to simplify the expression systematically.
• Radicals represent an operation on variables that need to be simplified independently before combining terms.

By arranging terms, identifying operations like addition and subtraction, and understanding coefficients, algebraic expressions can be tackled more effectively. This expression first needed each radical simplified, leading to easily combining terms. Learning to work with algebraic expressions equips you with skills vital for algebra, making complex operations manageable and solutions attainable.