Problem 79 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The simplified form of the given radical expression is \(2\sqrt{5}\)

Step 1: Identify the terms

The first step is to identify the terms in the radical expression. Here, \(\frac{1}{2}\) is a constant term outside the radical and 80 under the square root.

Step 2: Prime factorization

Use prime factorization to break down 80 into its base prime numbers. 80 can be written as \(2^4*5\).

Step 3: Simplify radical

Using properties of radicals, you can factor out the perfect square from under the square root. Here, we have \(2^4\), which is a perfect square. This can be simplified to \(2*2 = 4\). So, the square root of 80 simplifies to \(4\sqrt{5}\).

Step 4: Multiply with the constant

To find the final simplified form, multiply the constant term \(\frac{1}{2}\) with the simplified square root form \(4\sqrt{5}\).

Step 5: Final Simplification

\(\frac{1}{2}\) multiplied by \(4\sqrt{5}\) gives \(2\sqrt{5}\), which is the simplified radical expression.

Key Concepts

Prime Factorization

Prime factorization is a method of breaking down a number into the set of prime numbers that multiply together to make it. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves, like 2, 3, 5, 7, etc.
For example, if we take the number 80, we start by dividing by the smallest prime number, which is 2:

• 80 ÷ 2 = 40
• 40 ÷ 2 = 20
• 20 ÷ 2 = 10
• 10 ÷ 2 = 5

Since 5 is also a prime number, we stop here. Thus, 80 can be expressed as the product of its prime factors: \( 2^4 \times 5 \). This method is crucial for simplifying radicals, as it helps us identify perfect squares within a number.

Properties of Radicals

Understanding properties of radicals is essential for simplifying radical expressions. Radicals have special properties that allow us to manipulate them like the product and quotient rules.

• Product Rule: \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \)
• Quotient Rule: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \)

These properties mean we can break down or combine radical expressions, making them easier to handle.
For the expression \( \sqrt{80} \), we can separate 80 into its prime factors \( 2^4 \times 5 \), then use the product rule to simplify it as \( \sqrt{(2^4) \times 5} = \sqrt{2^4} \times \sqrt{5} = 4\sqrt{5} \). Recognizing and using these rules can simplify many complex looking expressions.

Perfect Square

A perfect square is a number that can be expressed as the square of an integer (whole number). Recognizing perfect squares is invaluable when simplifying radials because they help to eliminate the square root operator.
Common perfect squares include: 1, 4, 9, 16, 25, 36, and so on.

• For example, \( 16 = 4^2 \). Thus, \( \sqrt{16} = 4 \).
• In our expression \( \sqrt{80} \), we observe \( 2^4 = 16 \), which is a perfect square.

Identifying and extracting perfect squares like \( \sqrt{2^4} = 4 \) from under the radical simplifies \( \sqrt{80} \) to \( 4\sqrt{5} \). Through practice, recognizing these will become second nature, simplifying radical expressions more easily.

Constants in Expressions

Constants are fixed numbers that do not change in value. In the expression \( \frac{1}{2} \sqrt{80} \), \( \frac{1}{2} \) is a constant located outside the radical. It may look intimidating, but it multiplies with whatever is simplified from inside the radical.

To simplify the entire expression, we first simplify the radical part: \( \sqrt{80} = 4\sqrt{5} \). We then multiply this result by the constant \( \frac{1}{2} \). So

• \( \frac{1}{2} \times 4\sqrt{5} = 2\sqrt{5} \).

Constants simply "scale" expressions. Their simplicity helps reduce expressions to their simplest form. It's always wise to handle the radical separately first, then apply the constant to streamline the process.