Question

Simplify the radical expression. $$\frac{1}{2} \sqrt{80}$$

Short answer

The simplified form is 2\(\sqrt{5}\)

Step-by-step solution

Breaking Down the Square Root

The first step is to break down the square root of 80 into its prime factors. As 80 = 2 * 2 * 2 * 2 * 5, we have the square root of 80 equals to the square root of (2 * 2) * (2 * 2) * 5, thus it simplifies to 2*2*sqrt(5), which is 4*sqrt(5).

Multiply the Fraction with Simplified Square Root

The next step is to multiply the simplified square root with the fraction \(\frac{1}{2}\) which results in \(\frac{1}{2} * 4\sqrt{5}\). When multiplying, the 4 in the square root simplifies with the 2 in the denominator of the fraction to get 2.\(\sqrt{5}\).

Write down the Final Expression

The final simplification produces the radical expression 2\(\sqrt{5}\)

Key concepts

Square Root Simplification

Understanding the square root simplification is crucial in algebra and pre-calculus. A square root, denoted by the symbol \(\root{4}{\textstyle}\), represents a number which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \(3 \times 3\) equals 9.

To simplify a square root, we aim to find the prime factors of the number and identify pairs for removal. The number under the square root sign is known as the radicand. If the radicand is a perfect square, like 16 or 81, the simplification is straightforward since these are squares of 4 and 9, respectively. However, when dealing with a non-perfect square like 80, we need to break it down into its prime factors and then look for pairs, since a pair of primes under the square root can be simplified out of the radical.

In our exercise, \(\sqrt{80}\) can be simplified by finding the prime factors: 2, 2, 2, 2, and 5. Grouping the prime factors into pairs, we find that there are two pairs of 2's. Each pair comes out of the radical as a single 2, which when multiplied together outside the radical gives us 4, leaving us with 4\(\sqrt{5}\) once simplified.

Prime Factorization

Prime factorization is the method of expressing a number as the product of its prime factors. Primes are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. For example, the prime factors of 15 are 3 and 5. Why does this matter? In simplifying radical expressions, we often utilize prime factorization to break down composite numbers into a form that is easier to work with.

To perform prime factorization, you can use a factor tree or systematically divide by prime numbers. Let's consider the number 80 from our exercise. If we start dividing by the smallest prime, 2, we see that 80 is divisible several times: \(80 ÷ 2 = 40\), \(40 ÷ 2 = 20\), \(20 ÷ 2 = 10\), and \(10 ÷ 2 = 5\), where 5 is also a prime. So, 80 can be written as \(2 × 2 × 2 × 2 × 5\), which is the basis for simplifying square roots as we've previously discussed.

Radical Expression Simplification

Radical expression simplification involves reducing expressions under the radical sign to their simplest form. This could involve removing perfect square factors from underneath the radical or rationalizing a denominator with a radical in it. A radical expression is any mathematical expression containing a square root or higher roots.

When working with radical expressions, we follow certain rules and steps. After prime factorizing the radicand as we've discussed, we simplify the square root by bringing out any prime pairs as a single number. If there's a coefficient outside the square root, such as the \(\frac{1}{2}\) in our exercise, it's important to multiply it by the number that comes out of the radical. In this case, \(\frac{1}{2} * 4\sqrt{5}\) simplifies since \(\frac{1}{2} * 4\) equals 2, leaving us with a final simplified expression of \(2\sqrt{5}\).

Through these steps, we reach a simplified radical expression that is easier to work with and understand.