Problem 80 - Algebra and Trigonometry

Question

Simplify: \(\sqrt{108}-\sqrt{147}+\sqrt{363}\)

Step-by-Step Solution

Answer

10\sqrt{3}

Step 1 - Simplify each square root

Simplify each expression under the square root: \ \( \begin{aligned} \ \ \sqrt{108} &= \sqrt{36 \times 3} = \sqrt{36} \cdot \sqrt{3} = 6\sqrt{3} \ \ \sqrt{147} &= \sqrt{49 \times 3} = \sqrt{49} \cdot \sqrt{3} = 7\sqrt{3} \ \ \sqrt{363} &= \sqrt{121 \times 3} = \sqrt{121} \cdot \sqrt{3} = 11\sqrt{3} \end{aligned} \)

Step 2 - Substitute simplified forms back

Substitute the simplified square roots back into the original expression: \ \( 6\sqrt{3} - 7\sqrt{3} + 11\sqrt{3} \)

Step 3 - Combine like terms

Combine like terms: \ \( 6\sqrt{3} - 7\sqrt{3} + 11\sqrt{3} \). This equals \ \((-1\sqrt{3} + 11\sqrt{3} = 10\sqrt{3})\).

Key Concepts

Radicals

Radicals are expressions that include the symbol \(\text{√}\). This symbol indicates the square root, which represents a number that, when multiplied by itself, gives the original number inside the radical. For example, \(\text{√4}\) equals 2 because 2 × 2 equals 4. Radicals can simplify complex expressions when you know their properties.
In mathematical problems, especially those involving square roots, you often come across the term ‘radical.’ Understanding how to manipulate and simplify radicals is crucial. For instance, \(\text{√108}\) is a radical that can be simplified for easier calculation.

Like Terms

Like terms are terms that have identical variables raised to the same power. In the context of radicals, it specifically refers to radicals that contain the same number under the square root. For example, \(\text{6√3}\), \(\text{7√3}\), and \(\text{11√3}\) are like terms because they all have \(\text{√3}\).
Combining like terms means performing operations like addition or subtraction on these terms. Using the given example, simplifying the expression \(\text{6√3} - \text{7√3} + \text{11√3}\) entails adding and subtracting coefficients while keeping the radical part \(\text{√3}\) constant. This results in \(\text{10√3}\).

Simplification

Simplification involves reducing an expression to its simplest form. This makes it easier to understand and work with. When dealing with radicals, simplification often entails breaking down the number inside the square root to its prime factors and separating them into pairs.
For instance, to simplify \(\text{√108}\):

• Prime factorize 108: 108 = 36 × 3
• Recognize that 36 is a perfect square (6 × 6)
• So, \(\text{√108}\) = \(\text{√36}\) × \(\text{√3}\) = 6\(\text{√3}\)

This process is applied similarly to the other radicals in the exercise by finding their prime factors.

Square Roots Decomposition

Decomposition of square roots means breaking down the number under the radical into factors that include perfect squares. This decomposition helps in simplifying radicals.
For example, consider the number 147:

• Break down 147 into factors: 147 = 49 × 3
• Recognize that 49 is a perfect square (7 × 7)
• So, \(\text{√147}\) = \(\text{√49}\) × \(\text{√3}\) = 7\(\text{√3}\)

By breaking numbers into their prime factors, and recognizing and extracting perfect squares, decomposition makes it easier to handle and combine these radicals effectively in mathematical expressions.