Problem 78 - Algebra 1

Question

Simplify the radical expression. $$\sqrt{50}$$

Step-by-Step Solution

Answer

The simplified radical expression of \(\sqrt{50}\) is \(5\sqrt{2}\).

Step 1: Prime Factorization of Number

Decompose the number 50 into its prime factors. The prime factorization of 50 is \(2 \times 5 \times 5\) or \(2 \times 5^2\).

Step 2: Pairing the Prime Factors

Pair up the prime factors. In this case \(5^2\) makes a pair.

Step 3: Simplifying the Expression

Take out the pairs from under the square root. For each pair, one number can come out from under the radical as it makes a square. So, the number \(5\) comes out and the number \(2\) left under the radical because it's not part of a pair. So the square root becomes \(5\sqrt{2}\).

Key Concepts

Prime Factorization

Prime factorization is a process of breaking down a number into the prime numbers that, when multiplied together, give the original number. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. For the number 50, we can start by dividing by the smallest prime number, which is 2. This gives us:

• 50 divided by 2 equals 25.

Now, look at 25. Since 25 is not divisible by 2, try the next smallest prime number, which is 5:

• 25 divided by 5 equals 5.

Once you reach a quotient of 5, which is itself a prime number, you can stop. Therefore, the prime factorization of 50 is expressed as \( 2 \times 5 \times 5 \) or \( 2 \times 5^2 \).
This breakdown is essential because it is the first step in simplifying radical expressions.

Square Root

A square root is a value that, when multiplied by itself, gives the original number. The square root of a number is represented using the radical symbol \( \sqrt{} \). In simple terms, it reverses the process of squaring a number. For example, the square of 7 is 49 (since \( 7 \times 7 = 49 \)), so the square root of 49 is 7.
The square root function is essential in simplifying radical expressions because it allows us to reduce complex numbers to simpler forms. When simplifying \( \sqrt{50} \), you look for perfect square pairs within its prime factorization to make simplification possible.

Pairing Prime Factors

After obtaining the prime factorization of a number, the next step in simplifying a square root expression is to pair the prime factors. This involves looking for pairs of the same prime number. Each complete pair can be removed from under the square root and placed outside the radical symbol, simplified as a single number. In the case of 50, from its prime factorization \( 2 \times 5^2 \), we notice the pair \( 5^2 \). This is because 5 appears twice, making a perfect square pair.

• For each pair found, one number of the pair will "exit" the root.
• In this case, since there is a pair of 5's, a single 5 comes out of the square root.

The remaining number, 2 in this example, stays under the square root because it doesn't form a pair. Thus, we simplify \( \sqrt{50} \) to \( 5\sqrt{2} \).
This final form shows how complex numbers under a square root can be reduced into a simpler expression by understanding and applying the technique of pairing prime factors.