Problem 30 - Elementary and Intermediate Algebra

Question

Simplify each radical expression. All variables represent positive real numbers. $$ \sqrt{363} $$

Step-by-Step Solution

Answer

\(\sqrt{363} = 11\sqrt{3}\)

Step 1: Factorize the Radicand

We need to factor 363 into its prime factors. Start by dividing by the smallest prime number, which is 3. \[ 363 \div 3 = 121 \] Now factor 121, which is \(11 \times 11\), or \(11^2\). Thus, the factorization of 363 is \(3 \times 11^2\).

Step 2: Apply the Radical Rule

The square root of a number can be expressed by removing any squares from under the radical sign. Apply the rule \(\sqrt{a^2 \cdot b} = a\sqrt{b}\). Here, \(363 = 3 \times 11^2\), so \[ \sqrt{363} = \sqrt{3 \times 11^2} = 11\sqrt{3} \] by removing \(11^2\) from under the radical, because \(\sqrt{11^2} = 11\).

Key Concepts

Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime factors. These are numbers greater than 1 with no divisors other than 1 and itself. To find prime factors, start with the smallest prime, which is 2, and divide the number if it's even. Otherwise, move to the next smallest prime numbers like 3, 5, 7, and so forth.

For example, to factor 363, first check if it can be divided by 2. It's an odd number, so it isn't divisible by 2. We then try 3, the next smallest prime. Since 363 divided by 3 equals 121, we know 3 is a factor. Now, factor 121 by dividing by the smallest primes as well. In this case, 121 equals \(11 \times 11\), or \(11^2\). Thus, 363 is expressed as \(3 \times 11^2\).

This prime factorization helps in simplifying roots, especially square roots. The idea is to pair primes in power form, like \(11^2\), for easier computation when under a radical.

Square Roots

Square roots involve finding a number which, when multiplied by itself, gives the original number. It's represented by the radical symbol \( \sqrt{} \). Simplifying square roots often requires removing squares from under the radical. This is where our prime factorization comes into play.

To simplify \( \sqrt{363} \), first factor the number into primes: \(3 \times 11^2\). Notice \(11^2\), which is a perfect square, allows us to simplify the expression. According to radical rules, \(\sqrt{a^2 \times b} = a \sqrt{b}\). Apply this rule to get \(11 \sqrt{3}\).

This form shows 11 came out from under the radical because it's a square, and we leave the other factor, 3, under the radical as it doesn't form a perfect square.

Algebraic Expressions

An algebraic expression involves numbers, variables, and operations like addition or multiplication. When we simplify an expression involving radicals, such as square roots, it's part of algebraic manipulations.

With variables, applying similar rules as with numbers is essential. Ensure any variable represents a positive real number to avoid complications associated with negative square roots. By using factoring and radical rules, extend these concepts to variables too.

For example, if you have \( \sqrt{x^2y} \), consider \(x\) as a positive real number. You can simplify to \(x\sqrt{y}\) since \( \sqrt{x^2} = x \). Knowing how to extend these simplifications helps in simplifying complex algebraic expressions, making problem-solving more straightforward.