Problem 80 - Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt{80 c} $$

Step-by-Step Solution

Answer

The simplified form is \( 4 \sqrt{5c} \).

Step 1: Identify the Prime Factorization

To simplify the square root, start by finding the prime factorization of the constant inside the square root. The number 80 can be broken down into prime factors: \( 80 = 2^4 \times 5 \).

Step 2: Apply the Product Property of Square Roots

Use the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). Rewriting the expression, we have \( \sqrt{80c} = \sqrt{2^4 \times 5 \times c} \).

Step 3: Simplify the Square Roots

The square root of \( 2^4 \) is \( 2^2 \) because \( 2^2 \times 2^2 = 2^4 \). Therefore, \( \sqrt{2^4} = 4 \). The expression now is \( 4 \sqrt{5c} \).

Step 4: Combine and Write the Simplified Expression

Combine the simplification into one final expression. The final simplified form is \( 4 \sqrt{5c} \).

Key Concepts

Prime Factorization

Prime factorization is a method used to break down a number into its most basic building blocks, which are prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
For example, the number 80 can be broken down into prime factors as follows:

• 80 can be divided by 2 (which is prime) to give 40.
• 40 can again be divided by 2, resulting in 20.
• Continuing with 20, divide by 2 to get 10.
• Next, divide 10 by 2 to end up with 5.
• Now, 5 is already a prime number.

The prime factorization of 80 is therefore written as: \[ 80 = 2^4 \times 5 \]When simplifying square roots, knowing the prime factorization helps us see which factors can be paired to be taken out of the square root.

Product Property of Square Roots

The product property of square roots allows us to separate or combine square roots based on multiplication. This property states:\( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \)This is a very handy rule for square root simplification. If we have a complex square root, such as \( \sqrt{80c} \), we can express this using the product property as:\[ \sqrt{80c} = \sqrt{80} \times \sqrt{c} \]But even further, since 80 is not a prime, we can use its factorization:\[ \sqrt{80} = \sqrt{2^4 \times 5} \]This allows us to handle different parts of the radical separately, making it easier to simplify. Essentially, it helps us break the problem down into smaller chunks, which are easier to manage.

Simplifying Square Roots

Simplifying square roots is the process of rewriting the square root in its simplest form. Once we apply prime factorization and the product property of square roots, we're ready to simplify the expression.Returning to our example, \( \sqrt{80c} \), we break it down:

• From prime factorization, we know \( 80 = 2^4 \times 5 \).
• We use the product property to separate: \( \sqrt{2^4 \times 5 \times c} \).
• The next step is to simplify \( \sqrt{2^4} \).

Since \( 2^4 = (2^2) \times (2^2) = 4 \times 4 = 16 \), we have \( 2^2 = 4 \).Thus, \( \sqrt{2^4} = 4 \).Finally, by substituting back, the expression becomes:\[ 4 \times \sqrt{5c} \]This is the simplified form of the original expression \( \sqrt{80c} \). By understanding this process, students can simplify any square root expression more confidently.