Problem 69 - Algebra 1: Concepts and Skills

Question

Write the radical expression in simplest form. $$ -\frac{1}{2} \sqrt{360} $$

Step-by-Step Solution

Answer

The simplified form of \(-\frac{1}{2} \sqrt{360}\) is \(-3\sqrt{10}\).

Step 1: Prime Factorization

Let's first write prime factorization of 360. We can write it as \(2^3 * 3^2 * 5\).

Step 2: Find Perfect Square Factors and Simplify

The square root of a perfect square p is \(p^{1/2}\). From the prime factorization we can see a perfect square, \(2^2\) and \(3^2\). As square root of \(2^2\) is 2 and square root of \(3^2\) is 3, we can simplify \(\sqrt{360}\) to be \(2 * 3 * \sqrt{2*5}\) or \(6\sqrt{10}\).

Step 3: Apply the Negative Coefficient

Lastly, we should not forget about the negative half in front of the square root. We multiply the result from step 2, \(6\sqrt{10}\), with -1/2. Thus, we would have \(-\frac{1}{2}*6\sqrt{10} = -3\sqrt{10}\).

Key Concepts

Prime Factorization

Prime factorization is a method where we express a number as the product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. This method is helpful in simplifying expressions, especially when dealing with square roots. To find the prime factors of a number, you can start by dividing it by the smallest prime number, 2.

• If the number is divisible by 2, you write down 2 as a factor and divide the number by 2. Repeat this process until the number is no longer divisible by 2.
• Next, move to the next smallest prime number, which is 3, and repeat the process.
• Continue this step-by-step division using successive prime numbers like 5, 7, 11, and so on, until the remainder is 1.

For the number 360 in our exercise, the factorization ends with the prime factors: \[360 = 2^3 \times 3^2 \times 5.\]Using prime factorization simplifies calculations by breaking numbers into manageable parts. It especially aids in identifying perfect squares, a prominent step when simplifying radical expressions.

Perfect Squares

Perfect squares are numbers that can be written as an integer raised to the power of two. Recognizing perfect squares is particularly important when you simplify square roots because they allow us to take values out of the square root for a cleaner, simpler expression.
The key to simplifying radical expressions is finding and extracting perfect squares from the number under the square root sign.

• For example, if you have a perfect square like \(4\), since it's equal to \(2^2\), its square root is \(2\).
• Similarly, \(9 = 3^2\), so \(\sqrt{9} = 3\).
• In our case, the perfect squares involved in \(360\)'s prime factorization are \(2^2\) and \(3^2\).

By identifying these perfect squares, we are able to pull their square roots out of the radical, leaving us with simplified expressions such as \(6\sqrt{10}\). Understanding perfect squares helps in quickly streamlining the computation process, reducing complex radical forms to more straightforward calculations.

Square Roots

Square roots involve finding a value that, when multiplied by itself, yields the original number. This operation is the reverse of squaring a number. In the context of simplifying expressions, the square root helps retrieve simpler numbers from under the radical sign. Here’s a simple breakdown of square roots:

• The square root of a number \(x\), denoted as \(\sqrt{x}\), is a value that satisfies \(n^2 = x\).
• If \(x\) is a perfect square, its square root \(n\) is an integer. For \(x = 16\), since \(16 = 4^2\), \(\sqrt{16} = 4\).
• For non-perfect squares, like \(10\) in \(\sqrt{10}\), the square root is not an integer, and the expression remains in the radical form.

In our exercise, after simplifying \(\sqrt{360}\) using perfect squares and their roots, we are left with \(6\sqrt{10}\). This is further multiplied by the given negative coefficient in the expression, leading us to the final answer, \(-3\sqrt{10}\). Square roots with prime factorization ultimately enable us to rectify expressions into compact, efficient forms, paving the way for better numerical understanding.