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Chapter 8: Problem 39

Express each radical in simplified form. $$ \sqrt{288} $$

Short Answer

Expert verified
The simplified form of \(\sqrt{288}\) is 12\sqrt{2}.

Step by step solution

01

Prime Factorization

Find the prime factorization of 288. Start by dividing by the smallest prime number, which is 2: 288 梅 2 = 144144 梅 2 = 7272 梅 2 = 3636 梅 2 = 1818 梅 2 = 99 梅 3 = 33 梅 3 = 1So the prime factorization of 288 is: 288 = 2^5 脳 3^2.
02

Apply Square Root Property

Express the square root using the prime factors: \( \sqrt{288} = \sqrt{2^5 \times 3^2} \).
03

Simplify the Square Root

Identify pairs of prime factors under the square root. Each pair of the same number can be taken out of the square root. \( \sqrt{2^5 \times 3^2} = \sqrt{(2^2)^2 \times 2 \times 3^2} = 2^2 \times 3 \times \sqrt{2} = 4 \times 3 \times \sqrt{2} = 12 \sqrt{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Factorization
Prime factorization is a method used to simplify complex numbers by breaking them down into their prime components. A prime number is a natural number greater than 1 that has no divisors other than 1 and itself. To perform prime factorization, start by dividing the number by the smallest prime, which is 2, and continue dividing the quotient by prime numbers until only 1 remains. For example, to factorize 288, we repeatedly divide by 2 until we cannot anymore, then proceed with the next smallest prime, 3. This process results in: 288 = 2^5 脳 3^2. Prime factorization simplifies working with radicals as it allows us to break down the square root into manageable parts.
Square Root Property
The square root property helps in simplifying expressions involving square roots. When we split the number inside the square root according to its prime factors, it becomes easier to identify which parts can be taken out of the root. For instance, consider the prime factorization of 288: 288 = 2^5 脳 3^2. Using the square root property, we can rewrite the square root as: $$ \sqrt{288} = \sqrt{2^5 \times 3^2} \text. By identifying and labeling pairs of the same prime number, we can safely extract these pairs from under the square root, which brings us to the final step in simplifying the radical.
Radical Simplification
Radical simplification involves reducing a square root to its simplest form. This is done by taking advantage of the properties of square roots and prime factorization. Starting from the factorized form of 288 under the square root, $$ \sqrt{2^5 \times 3^2} \text. Look for pairs of prime factors: (\(2^2\))^2 \times 2 \times 3^2. Each pair can be taken out of the square root as a single number. Hence: $$ 2^2 = 4 and \text, 3^2 = 9: $$ \sqrt{(2^2)^2 \times 2 \times 3^2} = 4 \times 3 \times \sqrt{2} = 12\sqrt{2}. This makes our simplified expression: 12 \sqrt{2}. Simplifying radicals in this manner makes it easier to handle and understand complex algebraic expressions.

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