Problem 18 - Prealgebra

Question

Simplify each square root, then combine if possible. Assume all variables represent positive numbers. $$5 \sqrt{40}-2 \sqrt{90}+3 \sqrt{10}$$

Step-by-Step Solution

Answer

The simplified expression is \( 7 \sqrt{10} \).

Step 1: Simplify \( \sqrt{40} \)

To simplify \( \sqrt{40} \), find the largest perfect square factor of 40. The factorization of 40 is \( 4 \times 10 \), where 4 is a perfect square. Thus, \( \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2 \sqrt{10} \).

Step 2: Simplify \( \sqrt{90} \)

Find the largest perfect square factor of 90. The factorization is \( 9 \times 10 \), where 9 is a perfect square. Thus, \( \sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3 \sqrt{10} \).

Step 3: Combine the Simplified Terms

Now substitute the simplified roots back into the expression: \[5 \sqrt{40} - 2 \sqrt{90} + 3 \sqrt{10} = 5 (2 \sqrt{10}) - 2(3 \sqrt{10}) + 3 \sqrt{10} \]This simplifies to:\[10 \sqrt{10} - 6 \sqrt{10} + 3 \sqrt{10} \]

Step 4: Add and Subtract Like Terms

Combine like terms by performing the operations involving \( \sqrt{10} \):\[ (10 - 6 + 3) \sqrt{10} = 7 \sqrt{10} \]

Key Concepts

Perfect Square Factors

To effectively simplify square roots, identifying perfect square factors is crucial. A perfect square is a number that can be expressed as the product of an integer with itself, such as 4, 9, 16, or 25.
In the exercise, we start with numbers like 40 and 90. To break these down, we look for perfect square factors.

• For 40, the factorization is 4 (a perfect square) and 10, yielding: \( \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2 \sqrt{10} \).
• For 90, the factorization involves 9 (another perfect square) and 10: \( \sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3 \sqrt{10} \).

Recognizing and using perfect square factors simplifies square roots and prepares expressions for further operations.

Like Terms

In algebra, like terms are terms that have identical variable parts. For radicals, like terms have the same radicand, the number under the square root.

• From the simplified expression, we find: \(10 \sqrt{10} - 6 \sqrt{10} + 3 \sqrt{10}\).
• Since all terms involve \(\sqrt{10}\), they are like terms and can be readily combined.

Combining like terms streamlines the expression while keeping operations within the same category, ultimately resulting in a simplified solution.

Simplifying Radicals

When simplifying radicals, the goal is to present the simplest form, often breaking down numbers under the square root into smaller factors.

• Initially, reduce numbers to include perfect square factors, providing a straightforward path for radical simplification.
• For example, by substituting simplified forms back into the expression, e.g., \(5 (2 \sqrt{10}) - 2 (3 \sqrt{10}) + 3 \sqrt{10}\), you can consolidate calculations.
• Recognizing perfect squares quickly is crucial in simplifying and solving more complex expressions.

This lucid process aids problem-solving, making complex expressions easier to handle and understand.