Problem 8 - Algebra 1: Concepts and Skills

Question

Match the radical expression with its simplest form. $$ \sqrt{54} $$ A. \(3 \sqrt{6}\) B. \(5 \sqrt{3}\) C. \(7 \sqrt{2}\) D. \(3 \sqrt{5}\)

Step-by-Step Solution

Answer

The simplest form of \(\sqrt{54}\) is A. \(3 \sqrt{6}\)

Step 1: Prime Factorization

Break down the number 54 into its prime factors. Write 54 as \(2 \times 3^3\).

Step 2: Recognizing Perfect Square Factors

Identify the perfect square in the list of prime factors. 9, which is \(3^2\), is a perfect square. So rewrite the prime factors, grouping the repeated factor of 3 as \((3^2) \times 2 \times 3\).

Step 3: Simplifying the Radical

Now, apply the square root separately. The square root of \(3^2\) is 3. Therefore, \(\sqrt{9 \times 2 \times 3}\) simplifies to \(3 \sqrt{2 \times 3}\), which is equal to \(3 \sqrt{6}\).

Key Concepts

Prime Factorization

To simplify a radical expression, like \(\sqrt{54}\), the first step involves breaking down the number into its prime factors. This is called prime factorization. Each number can be expressed as a product of primes, which are numbers greater than 1 that have no divisors other than 1 and themselves. Let's take a closer look:

• Start with 54.
• It's divisible by 2 (an even number), which gives us \(54 \div 2 = 27\).
• Next, 27 can be divided by 3 (since 27 is a multiple of 3), resulting in \(27 \div 3 = 9\).
• Finally, 9 can be divided by 3 again, giving \(9 \div 3 = 3\).
• The last 3 is prime, and we stop here.

So, the prime factorization of 54 is \(2 \times 3^3\). Recognizing these prime numbers lets us further simplify wave functions much easier on the upcoming steps.

Perfect Squares

In simplifying radicals, finding and using perfect squares among the factors is essential. Perfect squares are numbers that can be expressed as the square of an integer. For instance, in our prime factors from the last step \(2 \times 3^3\), you'll notice \(3^3\) can be rearranged to highlight a perfect square:

• \(3^3 = (3^2) \times 3 = 9 \times 3\)

Here, \(3^2 = 9\) is a perfect square. Recognizing perfect squares in the factorization helps in simplifying radicals since you can easily calculate their square roots. Perfect squares streamline the simplification process significantly because they can be "pulled out" of the radical.

Square Roots

Understanding square roots is pivotal in simplifying radicals. The square root effectively "undoes" squaring. When you identify the perfect square during factorization, like \(9\) from \(3^2\), you can simplify the expression by removing it from under the square root:

• The square root of 9 is 3, simplifying the term \(\sqrt{9}\) to just 3.
• This conversion transforms \(\sqrt{54}\) to \(3 \sqrt{6}\).

The non-perfect square factors remain under the radical. Square roots provide an avenue to make expressions neater and more manageable by "collapsing" square pairs into their base integers. With this key step, simplifying radicals becomes much more intuitive. Remember, the aim is always to make computations easier and expressions clearer!