Problem 362 - Elementary Algebra

Question

In the following exercises, simplify and rationalize the denominator. $$ \sqrt{\frac{17}{192}} $$

Step-by-Step Solution

Answer

\frac{\root{51}}{24}

Step 1: Simplify the Radicand

Given the expression \(\root{\frac{17}{192}}\), start by simplifying the fraction under the square root. First, find the prime factorization of both the numerator and the denominator.17 is already a prime number.192 can be factored as follows: \192 = 2 \times 96 = 2 \times 2 \times 48 = 2 \times 2 \times 2 \times 24 = 2 \times 2 \times 2 \times 2 \times 12 = 2 \times 2 \times 2 \times 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3\.So, \192 = 2^6 \times 3\.Thus, \(\root{\frac{17}{192}} = \root{\frac{17}{2^6 \times 3}}\).

Step 2: Separate the Square Roots

Divide the square root of the fraction into a numerator and a denominator: \(\root{\frac{17}{2^6 \times 3}} = \frac{\root{17}}{\root{2^6 \times 3}}\).Simplify further: \(\frac{\root{17}}{\root{2^6} \times \root{3}}\).

Step 3: Calculate the Square Roots

Compute the square roots in the denominator: \(\root{2^6} = \root{64} = 8\).So the expression is now: \(\frac{\root{17}}{8 \times \root{3}} = \frac{\root{17}}{8\root{3}}\).

Step 4: Rationalize the Denominator

Rationalize the denominator by multiplying both the numerator and the denominator by \(\root{3}\) to cancel out the square root in the denominator: \(\frac{\root{17}}{8\root{3}} \times \frac{\root{3}}{\root{3}} = \frac{\root{17} \times \root{3}}{8 \times 3} = \frac{\root{51}}{24}\).

Key Concepts

Simplifying Radicals

Simplifying radicals can seem tricky at first, but it's straightforward once you get the hang of it. A radical expression involves a square root, cube root, or higher roots. When simplifying radicals, the goal is to make the expression as simple as possible, usually by factoring the radicand (the number under the root symbol) into its prime components and simplifying from there. For example, if we have \(\root{192}\), breaking it down into prime factors yields \(2^6 \times 3\). This step helps in further simplifying the expression by dealing with easily recognizable factors rather than large numbers.

Prime Factorization

Prime factorization is the process of breaking down a number into its prime number components. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves, such as 2, 3, 5, 7, and so on. To find the prime factorization of a number, you start with the smallest prime number (2) and divide the number by it, then move to the next smallest prime and continue this process until you're left with 1. For example, 192 can be factored as follows: \(192 = 2^6 \times 3\). Breaking down numbers into prime factors simplifies complex mathematical expressions and is particularly useful in simplifying radicals and rationalizing denominators.

Square Roots

A square root asks the question: what number, when multiplied by itself, gives the original number? For instance, \(\root{64} = 8\), because \(8 \times 8 = 64\). When dealing with square roots in fractions, it's often helpful to separate the numerator and the denominator to manage them individually. This helps simplify the expression more efficiently. For example, \(\root{\frac{17}{192}}\) can be broken down into \(\frac{\root{17}}{\root{192}}\), and then further simplified by working with each part. Understanding square roots is crucial to simplifying radicals and performing operations like rationalizing the denominator.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD). In the context of radicals, fraction simplification often requires rationalizing the denominator. Rationalizing the denominator involves removing any radical expressions in the denominator. For instance, in the fraction \(\frac{\root{17}}{8\root{3}}\), we can multiply the numerator and the denominator by \(\root{3}\) to eliminate the square root from the denominator. This results in \(\frac{\root{17} \times \root{3}}{8 \times 3} = \frac{\root{51}}{24}\). Simplifying fractions makes the expressions easier to understand and work with, especially in mathematical problem-solving.