Problem 7 - Prealgebra

Question

Simplify each expression by taking as much out from under the radical as possible. You may assume that all variables represent positive numbers $$\sqrt{98}$$

Step-by-Step Solution

Answer

\(\sqrt{98} = 7\sqrt{2}\).

Step 1: Identify the prime factors

First, we need to find the prime factorization of 98. We start by dividing 98 by the smallest prime number, which is 2: \[98 \div 2 = 49\]49 is not divisible by 2, so we try the next smallest prime, which is 3, but 49 is not divisible by 3 either. Next, we try 5, and it's not divisible. Finally, we try 7, and we find:\[49 \div 7 = 7\]Thus, the prime factorization of 98 is \(2 \times 7 \times 7\) or \(2 \times 7^2\).

Step 2: Simplify the radical

Using the prime factorization from Step 1, we express the square root as:\[\sqrt{98} = \sqrt{2 \times 7^2}\]We can take the square root of the perfect square part (\(7^2\)) out of the radical:\[\sqrt{2 \times 7^2} = 7 \times \sqrt{2}\]So, the simplified form of \(\sqrt{98}\) is \(7\sqrt{2}\).

Key Concepts

Prime Factorization

Prime factorization is a technique used to express a number as a product of its prime numbers. Prime numbers are natural numbers greater than 1 that have no other divisors but 1 and themselves. To determine the prime factorization of a number, you can start with the smallest prime number and test its divisibility.

• For example, starting with 98, we first check if it can be divided by 2, the smallest prime. Since 98 is even, it is divisible by 2, giving us 49.


• Next, we move to the next smallest prime numbers - 3, 5, and finally 7 - checking each time if they divide the remaining quotient without a remainder. Here, 49 is not divisible by any of these until we try 7, and we see that 49 equals 7 times 7.

Therefore, the prime factorization of 98 is expressed as \(2 \times 7 \times 7\) or \(2 \times 7^2\). Knowing prime factorization sets the foundation for understanding and simplifying square roots.

Radicals

In mathematics, a radical is an expression that includes a square root, cube root, or higher root. Simplifying radicals involves expressing them in their simplest form, where any perfect square factors are extracted from under the radical sign.

• Taking our previous example \(\sqrt{98}\), we use its prime factorization \(2 \times 7^2\) to simplify the expression.


• We identify any number raised to the power of two (a perfect square) in the factorization - in this case, \(7^2\). This perfect square can be "taken out" of the radical because the square root of a perfect square is just the number itself.

The simplified form, therefore, is \(7\sqrt{2}\). This process not only makes the radical easier to read but also simpler for further calculations.

Perfect Squares

Perfect squares are numbers that are the square of an integer. Recognizing perfect squares is critical when simplifying square roots. In the context of our exercises, a perfect square is determined when examining the prime factorization of a number.

• For example, reconsider \(98 = 2 \times 7^2\). Here, \(7^2\) is a perfect square because it equals \(49\), the square of 7.


• When you find a perfect square under a radical, it can be extracted from the radical sign, turning it into its integer form outside the symbol. For \(\sqrt{98}\), the \(7^2\) becomes 7, simplifying the term to \(7\sqrt{2}\).

Identifying and extracting perfect squares is a useful technique in algebra that helps to simplify expressions and solve equations more efficiently.