Problem 43 - Algebra and Trigonometry

Question

Add or subtract terms whenever possible. $$ 3 \sqrt{8}-\sqrt{32}+3 \sqrt{72}-\sqrt{75} $$

Step-by-Step Solution

Answer

The answer to the addition and subtraction of the square root terms is: \(20 \sqrt{2} - 5\sqrt{3}\)

Step 1: Simplify each term

The first step is to simplify each term then express each root as multiples of square roots of whole numbers. Simplicity is achieved by factoring each term under the root symbol into a product involving a perfect square.\nSo we have: \(\sqrt{8} = \sqrt{4*2} = 2\sqrt{2}\), \(\sqrt{32} = \sqrt{16*2} = 4\sqrt{2}\), \(\sqrt{72} = \sqrt{36*2} = 6\sqrt{2}\) and \(\sqrt{75} = \sqrt{25*3} = 5\sqrt{3}\)

Step 2: Substitute the simplified terms

Now replace each of the original terms with their respective simplified forms in the original equation. We get \[3*2\sqrt{2} - 4\sqrt{2} + 3*6\sqrt{2} - 5\sqrt{3}\] which simplifies to \[6\sqrt{2} - 4\sqrt{2} + 18\sqrt{2} - 5\sqrt{3}\]

Step 3: Combine like terms

Combine terms that have the same square root. This gives us \( (6-4+18) \sqrt{2} - 5\sqrt{3}\). This simplifies to \(20 \sqrt{2} - 5\sqrt{3}\)

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics, especially when it comes to simplifying radicals. They involve finding a number that, when multiplied by itself, equals the original value. For example, the square root of 4 is 2 because

• 2 × 2 = 4

Understanding how square roots work is crucial for simplifying expressions involving radicals. In the given exercise, each term under the radical is factored into a combination of a perfect square and another number—a process that aids in simplifying the radical expression. For instance,

• \(\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}\), where 4 is a perfect square.
• This method is applied similarly to \(\sqrt{32} = 4\sqrt{2}\), \(\sqrt{72} = 6\sqrt{2}\), and \(\sqrt{75} = 5\sqrt{3}\).

By focusing on perfect square factors, we make it easier to simplify the expression further. This step paves the way for combining like terms.

Combining Like Terms

Combining like terms is an essential step in algebra that involves grouping together terms that have identical variable factors. This process simplifies expressions, making them easier to work with. In the exercise, like terms refer to terms that share the same square root. For example, \(6\sqrt{2}, -4\sqrt{2}, \) and \(18\sqrt{2}\) all can be combined since they share the \(\sqrt{2}\) factor.The key points of this process are:

• Identify terms with the same square root.
• Add or subtract their coefficients.
• Maintain the shared square root factor .

So, by adding and subtracting the coefficients of the like terms, we simplify the expression to \((6 - 4 + 18)\sqrt{2} - 5\sqrt{3}\), which reduces even further to \(20\sqrt{2} - 5\sqrt{3}\).This technique of combining like terms is fundamental in manipulating algebraic expressions efficiently.

Factoring Perfect Squares

Factoring perfect squares is another essential concept in simplifying radicals. This process involves identifying perfect square factors within a radical, which makes it easier to bring terms out of the radical for simplification. Perfect squares are numbers like 4, 9, 16, and 25, which have exact integers as their square roots.In practice:

• We factor the number under the square root into two products, one of which is a perfect square.
• For instance, \(\sqrt{8} \) was broken down into \(\sqrt{4 \times 2}\).
• The perfect square 4 is extracted as \(2\sqrt{2}\) since \(\sqrt{4} = 2\).
• This approach is similarly applied to \(\sqrt{32}=4\sqrt{2}\), \(\sqrt{72}=6\sqrt{2}\), and \(\sqrt{75}=5\sqrt{3}\).

Understanding how to efficiently factor perfect squares simplifies the process of manipulating and combining terms, leading to a clearer, more manageable expression.