Problem 16 - Elementary and Intermediate Algebra

Question

Simplify each expression. $$ 9 \sqrt{54} $$

Step-by-Step Solution

Answer

The simplified form is \( 27 \sqrt{6} \).

Step 1: Express the square root in terms of its factors

First, identify the prime factorization of the number inside the square root. The number 54 can be factored into prime numbers as follows: \( 54 = 2 imes 3^3 \). Therefore, we can rewrite \( \sqrt{54} \) as \( \sqrt{2 imes 3^3} \).

Step 2: Simplify using square root properties

Using the property of square roots that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \), we can simplify \( \sqrt{2 \times 3^3} \) to \( \sqrt{2} \times \sqrt{3^3} \). Remember that \( \sqrt{3^3} = \sqrt{27} = 3 \cdot \sqrt{3} \) since \( 3^2 = 9 \). Hence, \( \sqrt{54} = 3 \sqrt{6} \).

Step 3: Multiply the simplified square root by the outside factor

Returning to the original expression, we have \( 9 \sqrt{54} = 9 \times 3 \sqrt{6} \). Calculate \( 9 \times 3 = 27 \), so the expression becomes \( 27 \sqrt{6} \).

Step 4: Write the final simplified expression

The completely simplified form of the expression is \( 27 \sqrt{6} \).

Key Concepts

Understanding Prime Factorization

Prime factorization is a method used to express a number as a product of prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves, like 2, 3, and 5. In prime factorization, you break a number down into the smallest possible factors which are all prime. For example, the prime factorization of 54 is found by dividing 54 by the smallest prime number that goes into it evenly, which is 2 in this case. Therefore, 54 can first be expressed as \( 2 \times 27 \). Next, 27 can be broken down to \( 3 \times 9 \), and finally, 9 can be simplified to \( 3 \times 3 \). Thus, the prime factorization of 54 is \( 2 \times 3^3 \). This method helps significantly when simplifying expressions involving square roots as shown in the original problem, where knowing this factorization allows us to proceed with simplifying \( \sqrt{54} \).
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Mastering prime factorization provides a solid foundation for simplifying more complex expressions and understanding square roots better.

• A prime number is only divisible by 1 and itself.
• 54 factors into \( 2 \times 3^3 \).
• Always break down components to their prime factors for simplification.

Exploring Square Root Properties

Square roots simplify expressions involving radicals by using fundamental properties. These properties allow us to break down complicated square roots into simpler parts. One of the most important properties is \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). This means that the square root of a product can be split into the product of the square roots of each factor. Applying this to our problem: \( \sqrt{54} = \sqrt{2 \times 3^3} = \sqrt{2} \times \sqrt{27} \). Here, you can further explore \( \sqrt{27} \) as it is a perfect square of 9 times another factor. Since \( 27 = 3^3 \), this can be simplified to \( 3 \times \sqrt{3} \).
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Understanding these properties aids in efficiently simplifying expressions by focusing on combining or separating radicals, which becomes especially useful in exercises involving multiply by external factors.

• Use \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \) to simplify radicals.
• \( 3^3 \) leads to further simplification as \( 3 \times \sqrt{3} \).
• Eases handling expressions with multiple radical terms.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This process often involves finding and combining like terms or applying mathematical properties to remove unnecessary complexity. In the example problem, you begin with \( 9 \sqrt{54} \). Having already simplified \( \sqrt{54} \) to \( 3 \sqrt{6} \), the next step is to incorporate the external factor, which is 9 in this case. Multiply 9 by the simplified expression, giving \( 9 \times 3 \sqrt{6} = 27 \sqrt{6} \). This is a clear example of how multiple mathematical processes can simplify even more complicated expressions by strategically applying properties of numbers and operations.
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By focusing on simplification strategies and step-by-step reduction, complex mathematical expressions become clearer and more manageable.

• Simplification combines applying properties and reducing redundancies.
• Multiplying across simplified expressions gives final results, like \( 27 \sqrt{6} \).
• Leads to concise answers that are easier to interpret and use in further calculations.