Problem 32 - Algebra for College Students

Question

Change each radical to simplest radical form. \(-4 \sqrt{54}\)

Step-by-Step Solution

Answer

The simplest radical form of \(-4 \sqrt{54}\) is \(-12 \sqrt{6}\).

Step 1: Prime Factorize the Radicand

Identify factors of the number under the square root. The radicand is 54. Start by dividing 54 by the smallest prime numbers. 1. Divide 54 by 2 to get 27. 2. Divide 27 by 3 to get 9. 3. Divide 9 by 3 to get 3, and 3 is a prime number.

Step 2: Write the Radicand as a Product of Primes

Express 54 as a product of its prime factors. From the factorization, we have 54 as:\[ 54 = 2 \times 3^3 \]

Step 3: Simplify the Square Root

Apply the property of square roots, \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). Here, simplify \(\sqrt{54}\).\[ \sqrt{54} = \sqrt{2 \times 3^3} = \sqrt{2} \times \sqrt{3 \times 3^2} \]Since \(3^2 = 9\) and \(\sqrt{9} = 3\), we have:\[ \sqrt{54} = \sqrt{2} \times 3 \times \sqrt{3} = 3 \sqrt{6} \]

Step 4: Apply the Simplification to the Original Expression

Substitute the simplified radical back into the original expression.The expression is \(-4 \sqrt{54}\). Replace \(\sqrt{54}\) with \(3 \sqrt{6}\):\[ -4 \times 3 \sqrt{6} = -12 \sqrt{6} \]

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its simplest building blocks, the prime numbers. Prime numbers are numbers that can only be divided by themselves and 1, such as 2, 3, 5, 7, and so on. To prime factorize a number, start with the smallest prime number that can divide it evenly. Continue dividing the result until you are left with all prime numbers.
For instance, in the exercise above, the number 54 is our target. We began by dividing by 2, the smallest prime number. Here’s the breakdown:

• 54 divided by 2 equals 27.
• 27 divided by 3 equals 9.
• 9 divided by 3 equals 3, which is a prime number.

Thus, 54 is factorized into the prime numbers: 2 and three 3's, expressed as \( 54 = 2 \times 3^3 \). This prime factorization is essential for simplifying radicals.

Square Roots

A square root is a number which, when multiplied by itself, gives the original number. Think of a square root as the opposite of squaring a number. The symbol for square root is \( \sqrt{} \). It can be applied to whole numbers as well as more complicated expressions.
In our example, we are dealing with the square root of 54. However, to simplify it, we rewrite 54 using its prime factors: \( 54 = 2 \times 3^3 \). The next step is applying the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). This helps us split the square root into manageable parts, which is crucial for simplifying radicals.

Radicals

Radicals are expressions that involve root symbols, like a square root, cube root, etc. The goal of working with radicals is often to make them as simple as possible while still representing the same value.
In simplifying radicals, the prime factorization gives a pathway to identify perfect squares within the number. From our given exercise with \( \sqrt{54} \), we found \( 54 = 2 \times 3^3 \). By extracting the perfect square \( 3^2 = 9 \), we know \( \sqrt{9} = 3 \). This allows us to further break it down to \( 3 \sqrt{6} \) by taking the 3 out of the radical. This is a powerful technique that helps turn complex roots into simpler expressions.

Simplifying Expressions

Simplifying expressions involves reducing them to their most concise form. When dealing with radicals, this may involve both algebraic manipulation and the principles of arithmetic.
From our example, the expression \( -4 \sqrt{54} \) was simplified through prime factorization and root simplification to \( -12 \sqrt{6} \). Here's what happens:

• First, identify and simplify the radical \( \sqrt{54} \) to \( 3 \sqrt{6} \).
• Next, multiply by the coefficient outside the radical, -4, with 3 to get -12.
• Finally, write the fully simplified expression as \( -12 \sqrt{6} \).

This step-by-step process results in a cleaner, more understandable form suitable for mathematical calculations.