Problem 37 - Contemporary Precalculus

Question

Simplify the expression without using a calculator. $$5 \sqrt{20}-\sqrt{45}+2 \sqrt{80}$$

Step-by-Step Solution

Answer

Question: Simplify the given expression: \(5\sqrt{20} - \sqrt{45} + 2\sqrt{80}\) Answer: The simplified expression is \(35\sqrt{5}\).

Step 1: Simplify the square roots

Let's start by simplifying each square root in the given expression. The largest perfect square factors for 20, 45, and 80 are 4 (4 * 5), 9 (9 * 5), and 16 (16 * 5), respectively. So, we can rewrite the expression as: $$5\sqrt{4 \cdot 5} - \sqrt{9 \cdot 5} + 2\sqrt{16 \cdot 5}$$ Now, use the property of square roots: \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\): $$5(\sqrt{4} \cdot \sqrt{5}) - (\sqrt{9} \cdot \sqrt{5}) + 2(\sqrt{16} \cdot \sqrt{5})$$ Evaluate the square roots with perfect squares: $$5(2\sqrt{5} - 3\sqrt{5} + 2(4\sqrt{5}))$$

Step 2: Perform the operations

Combine the terms with the same square root: $$ 5(2\sqrt{5} - 3\sqrt{5} + 8\sqrt{5})$$ $$5((2-3+8)\sqrt{5})$$ Combine the numbers within the parentheses: $$5(7\sqrt{5})$$ Now, multiply the numbers outside the square root: $$35\sqrt{5}$$ The simplified expression is: $$35\sqrt{5}$$

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics, particularly in algebra and geometry. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, in this expression, we need to simplify terms like \(\sqrt{20}\), \(\sqrt{45}\), and \(\sqrt{80}\). To make simplification easier, we look for the largest perfect square factors within each of these numbers.

When simplifying square roots, the key principle is recognizing that \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\). This property allows us to separate a square root into more manageable parts.

Here are the steps taken from the problem:

• \(\sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5}\)
• \(\sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}\)
• \(\sqrt{80} = \sqrt{16 \cdot 5} = \sqrt{16} \cdot \sqrt{5} = 4\sqrt{5}\)

This simplification makes it easier to perform further operations by reducing the complexity of the original expression.

Perfect Squares

Perfect squares are numbers that can be expressed as some integer multiplied by itself. Recognizing perfect squares plays a crucial role in simplifying square roots.

Numbers like 4, 9, and 16, which are used in this exercise, are examples of perfect squares since:

• \(4 = 2^2\)
• \(9 = 3^2\)
• \(16 = 4^2\)

Knowing such numbers helps in breaking down larger numbers into smaller, factorable parts.

In the given expression, the presence of perfect squares under the square roots (e.g., \(4 \cdot 5\), \(9 \cdot 5\), and \(16 \cdot 5\)) allows us to simplify the radicals by taking out their square root part:

• \(\sqrt{4}\) becomes 2
• \(\sqrt{9}\) becomes 3
• \(\sqrt{16}\) becomes 4

This process reduces the square root expressions into their simplest form, making the equations in the problem easier to manage.

Algebraic Manipulation

Once we've simplified each square root using perfect squares, algebraic manipulation is used to combine terms. The expression we simplified initially was \(5\sqrt{4 \cdot 5} - \sqrt{9 \cdot 5} + 2\sqrt{16 \cdot 5}\), which simplifies to \(5(2\sqrt{5}) - (3\sqrt{5}) + 2(4\sqrt{5})\).

Algebraic manipulation involves performing operations on these simplified terms:

• Group the terms that have the same square root \(\sqrt{5}\).
• Apply distribution: \((5 \times 2)\) for the first term, \((-1 \times 3)\) for the second term, and \((2 \times 4)\) for the third term.
• Combine the coefficients with the like term \(\sqrt{5}\).
• This results in \((10 - 3 + 8)\sqrt{5}\) which simplifies to \(7\sqrt{5}\).
  
  Finally, multiply the existing outside coefficient by the combined term to get \(35\sqrt{5}\).

This method of grouping similar terms and simplifying the expression step-by-step provides a clear pathway to solving complex algebraic problems.