Problem 76 - Algebra 1: Concepts and Skills

Question

Simplify the radical expression. \(\sqrt{128}\)

Step-by-Step Solution

Answer

The simplified form of the radical expression \(\sqrt{128}\) is \(8\sqrt{2}\)

Step 1: Prime factorization of the number under the square root

First, write the number 128 as a product of its prime factors. Prime factors are basically the prime numbers (like 2, 3, 5 etc.) that can divide the number without leaving a remainder. Here, the prime factors of 128 are 2 as it can be written as \(128 = 2^7\)

Step 2: Apply the rule of radicals

Every time there's a pair of same factors under a square root, we can pull it out of the root sign. Here, we have 3 pairs of 2s, so we pull them out. This gives us \(2 * 2 * 2 * \sqrt{2} = 8\sqrt{2}\)

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, which are the prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
For instance, 2, 3, 5, 7, and 11 are examples of prime numbers. When we factorize a number, we express it as a product of prime numbers.

For the number 128, prime factorization looks like this:

• Divide 128 by 2 (the smallest prime number), to get 64.
• Divide 64 by 2 to get 32.
• Divide 32 by 2 to get 16.
• Divide 16 by 2 to get 8.
• Divide 8 by 2 to get 4.
• Divide 4 by 2 to finally get 2.

Through this step-by-step division, we conclude that 128 can be expressed as the product of prime factors: \[128 = 2^7\]Prime factorization is a crucial step in simplifying radical expressions, as it helps in simplifying the square root.

Square Root

Understanding the square root concept is essential when you are simplifying radical expressions. A square root of a number is a value that, when multiplied by itself, gives the original number. The square root is denoted by the symbol \(\sqrt{\cdot}\).

To make the concept clearer, consider this: the square root of 9 is 3 because 3 multiplied by 3 equals 9. Thus, we write \(\sqrt{9} = 3\). Similarly, when dealing with square roots within radical expressions like \(\sqrt{128}\), one goal is to simplify it by finding a simpler, equivalent expression. This often involves using prime factorization.

When you identify pairs of prime factors under the radical sign, you can "pull out" those pairs as single numbers outside the square root, since the square root operation is essentially asking "What number times itself gives the original number?" In the case of \(\sqrt{128} = 8\sqrt{2}\), we have identified three pairs of 2s, which equates to 8 when you "pull them out," leaving a single 2 under the radical sign.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and other mathematical expressions that involve a root sign \(\sqrt{\cdot}\). These expressions allow us to indicate numbers that can be multiplied by themselves a certain number of times to reach a given number.

Simplifying radical expressions is a process that involves reducing these expressions to their simplest form. One common method is through prime factorization and understanding how to "pull out" perfect squares from under the root symbol.

For instance, with \(\sqrt{128}\), breaking 128 down into its prime factors helps to simplify the expression. By recognizing 128 as \(2^7\), you see there are three pairs of 2s that can be "pulled out" of the radical, simplifying \(\sqrt{128} \) to \(8\sqrt{2}\). This means that while the expression under the square root contained redundancy in the form of repeated factors, simplifying it helped express it more neatly.
Words like "radical" and "expression" refer to the notation and presentation of these types of numbers. They may initially appear complex, but through simplification, they become easier to handle in equations, making calculations more straightforward in many mathematical and real-world applications.