Problem 27 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(\sqrt{\frac{360}{361}}\)

Step-by-Step Solution

Answer

\(\frac{6\sqrt{10}}{19}\)

Step 1: Recognize Perfect Squares

Identify whether the numbers under the square root are perfect squares. In this case, 360 is not a perfect square, while 361 is because \(361 = 19^2\).

Step 2: Simplify the Square Root of Perfect Numbers

Simplify the expression by taking the square root of the perfect square number. Since \(361 = 19^2\), \(\sqrt{361} = 19\). Thus, the expression becomes \( \frac{\sqrt{360}}{19} \).

Step 3: Simplify the Numerator

Determine if \(360\) can be simplified further inside the square root. Factor \(360\) into its prime factors: \(360 = 2^3 \times 3^2 \times 5\). Group the factors into pairs to simplify: \(\sqrt{2^2 \times 3^2 \times 2 \times 5}\).

Step 4: Extract Perfect Squares from the Square Root

Take the square root of the perfect squares from the factorization: \(\sqrt{2^2} = 2\) and \(\sqrt{3^2} = 3\). So, \(\sqrt{360} = 2 \times 3 \times \sqrt{10} = 6\sqrt{10}\).

Step 5: Compose the Simplified Expression

Replace the numerator \(\sqrt{360}\) with its simplified form. The expression becomes \(\frac{6\sqrt{10}}{19}\). This fraction cannot be simplified further since 6, 10, and 19 have no common factors.

Key Concepts

Perfect Squares

Understanding perfect squares is fundamental when simplifying expressions under a square root. A perfect square is a number that can be expressed as the product of an integer with itself. For example, 1, 4, 9, 16, and 25 are all perfect squares because they can be written as:

• 1 = 1 × 1
• 4 = 2 × 2
• 9 = 3 × 3
• 16 = 4 × 4
• 25 = 5 × 5

Recognizing perfect squares within a numerical expression helps to simplify it. In the context of our exercise, 361 is identified as a perfect square because it equals 19 squared, or 19 × 19. Recognizing this allows us to compute the square root of 361 easily, simplifying the problem significantly.

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, which are prime numbers. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. For example, the prime numbers less than 10 are 2, 3, 5, and 7.

To perform prime factorization, you can divide a number by the smallest prime number possible, then continue the process with the quotient. This is key in simplifying radicals because it helps identify the factors inside a square root.

In our example exercise, 360 can be broken down into its prime factors as follows:

• 360 ÷ 2 = 180
• 180 ÷ 2 = 90
• 90 ÷ 2 = 45
• 45 ÷ 3 = 15
• 15 ÷ 3 = 5
• 5 ÷ 5 = 1

So, the prime factorization of 360 is 2³ × 3² × 5. Identifying these factors allows you to simplify the square root by grouping numbers into pairs.

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2, because 2 × 2 = 4. The √ symbol is used to denote the square root.

Some numbers, like 361, are perfect squares, meaning their square root is a whole number (19 in this case). However, not all numbers are perfect squares, which is where simplifying becomes relevant.

When dealing with non-perfect squares like 360, we use its prime factorization to extract perfect squares. In our example, having already factorized 360 into 2³ × 3² × 5, we were able to group pairs of the same numbers:

• Extract the perfect square pair of primes, 2² , which is 4, and 3² , which is 9.
• Square roots of these are √4 = 2 and √9 = 3 .

This means √360 = 2 × 3 × √10 = 6 √10.

This method of simplifying square roots helps in reducing expressions to their simplest form, which is easier to work with in further calculations.