Problem 24 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\sqrt{98}\)

Step-by-Step Solution

Answer

The simplest radical form of \( \sqrt{98} \) is \( 7\sqrt{2} \).

Step 1: Identify the Prime Factors

Find the prime factors of the number 98. Start by dividing 98 by the smallest prime number, which is 2. Since 98 is even, divide it by 2 to get 49. Next, determine the prime factors of 49. It can be divided by 7, which is a prime number, resulting in 7. Thus, the prime factorization of 98 is \( 2 \times 7^2 \).

Step 2: Simplify the Radical

Using the prime factorization from Step 1, rewrite \( \sqrt{98} \) as \( \sqrt{2 \times 7^2} \). Since \( 7^2 = 49 \), it can be taken out of the square root as 7, leaving \( 7\sqrt{2} \).

Step 3: Write the Final Answer

Since \( \sqrt{98} \) simplifies to \( 7\sqrt{2} \), write this expression as the simplest radical form.

Key Concepts

Prime Factorization

Prime factorization is the process of expressing a number as a product of prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For instance, the prime numbers include 2, 3, 5, 7, and so on.

To find the prime factorization of a number, we repeatedly divide it by its smallest prime factor until we're left with prime numbers. Let's look at how we can break down 98 into prime factors:

• Start with the smallest prime number, 2. Since 98 is even, divide it by 2 to get 49.
• Next, verify 49. It's not divisible by 2, 3, or 5, but it is divisible by 7 (since 7 times 7 equals 49). Thus, 49 divides by 7 to give 7.
• As both 7s are prime numbers, the process stops here.

This results in the prime factorization of 98 as \(2 \times 7^2\). Prime factorization is a foundational tool for working with radicals and simplifying complex mathematical expressions.

Radicals

The concept of radicals is centered around the idea of expressing numbers in terms of their root values. A radical denotes the root of a number, commonly square roots, cube roots, etc. The symbol \(\sqrt{}\) represents the square root, but we can also have cube roots with \(\sqrt[3]{}\), and so on.

In the context of simplifying radicals, such as \(\sqrt{98}\), the goal is to identify parts of the number inside the radical that can be expressed as perfect squares or cubes.

• First, identify the prime factorization of the number inside the radical, like we did earlier with 98.
• Determine which factors can form pairs for square roots. For example, \(7^2\) forms a perfect square.
• Pull out the square-rooted factor outside the radical.

Thus, the expression \(\sqrt{98}\) simplifies to \(7\sqrt{2}\), thanks to the understanding of radicals and their properties.

Square Roots

Square roots are one of the most common types of roots found in mathematics. The square root of a number \(x\) is a value that, when multiplied by itself, gives \(x\). Symbolically, this is expressed as \(\sqrt{x}\).

To understand square roots, it is crucial to recognize how they relate to multiplications and perfect squares. Knowing how to find the square root involves recognizing numbers that can perfectly multiply into another, forming a square.

• Perfect squares are numbers like 1, 4, 9, 16, 25, where the square root is a whole number.
• Non-perfect squares, like 98, require simplification through prime factorization to extract as many perfect squares as possible.

Therefore, calculating a square root might initially seem challenging, but breaking down the number into its simplest form makes it easier. For instance, simplifying \(\sqrt{98}\) results in \(7\sqrt{2}\), achieving the simplest radical form through understanding and using square roots effectively.