Problem 16 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

Simplified, it is \( 27 \sqrt{6} \).

Step 1: Identify Prime Factors of 54

First, we need to find the prime factorization of 54 to simplify the square root. The prime factors of 54 are 2, 3, 3, and 3 because 54 can be expressed as \( 54 = 2 \times 3^3 \).

Step 2: Break Down the Square Root

Using the prime factors, write \( \sqrt{54} \) as \( \sqrt{2 \times 3^3} \). According to the properties of square roots, this expression can be broken down as \( \sqrt{2} \times \sqrt{3^3} \).

Step 3: Simplify the Square Root of a Perfect Square

The expression \( \sqrt{3^3} \) can be simplified. Since \( 3^3 = 9 \times 3 \), we have \( \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} \). We know \( \sqrt{9} = 3 \), so it becomes \( 3 \sqrt{3} \).

Step 4: Combine Simplified Parts

Now substitute back: \( \sqrt{54} = \sqrt{2} \times 3 \sqrt{3} \). Hence, \( 9 \sqrt{54} = 9 \times \sqrt{2} \times 3 \sqrt{3} \).

Step 5: Final Simplification

Multiply the numbers: \( 9 \times 3 = 27 \). Therefore, the expression becomes \( 27 \sqrt{6} \). Thus, \( 9 \sqrt{54} = 27 \sqrt{6} \).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its smallest prime factors — numbers that are divisible only by 1 and themselves. To perform prime factorization, you repeatedly divide the number by the smallest prime numbers (2, 3, 5, 7, 11, ...) until you are left with 1. For example, when finding the prime factors of 54, we divide as follows:

• 54 ÷ 2 = 27
• 27 ÷ 3 = 9
• 9 ÷ 3 = 3
• 3 ÷ 3 = 1

The complete prime factorization of 54 is hence written as \( 2 \times 3^3 \).
By understanding the prime factorization, you can simplify math expressions more easily, such as square roots, which often rely on finding perfect squares hidden within the factors.

Square Roots

A square root asks the question: what number multiplied by itself gives the original number? For instance, the square root of 9 is 3, because 3 times 3 equals 9.
When dealing with an expression like \( \sqrt{54} \), you want to reduce it by breaking it down using its prime factors into simpler parts. From our previous prime factorization, \( \sqrt{54} \) is written as \( \sqrt{2 \times 3^3} \).
By using the property of square roots that says \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \), you can simplify it further. If any of the factors form a perfect square, they can be simplified separately, aiding the overall simplification process.

Perfect Squares

Perfect squares are numbers that have whole numbers as their square roots. This means if you can express a number as a product of a whole number by itself (like 1, 4, 9, 16, 25, etc.), it is considered a perfect square.
In our simplification process, recognizing \( 3^3 = 9 \times 3 = 27 \) involves finding a perfect square, \( 9 \), as one part of it. Because \( 9 \) is a perfect square, \( \sqrt{9} \) simplifies to 3.
Understanding and identifying perfect squares within your expressions enable you to simplify calculations efficiently, paving the way for a clear and confident manipulation of algebraic expressions.