Problem 43 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\sqrt{\frac{75}{81}}\)

Step-by-Step Solution

Answer

\(\frac{5\sqrt{3}}{9}\)

Step 1: Simplify Inside the Radical

First, simplify the fraction under the square root sign. The fraction is \(\frac{75}{81}\). Note that both 75 and 81 can be divided by 3. Simplify it to \(\frac{25}{27}\).

Step 2: Prime Factorization

Perform prime factorization on the numerator and the denominator separately. The number 25 is \(5^2\), and 27 is \(3^3\).

Step 3: Rewrite the Radical

Rewrite \(\sqrt{\frac{25}{27}}\) as \(\frac{\sqrt{25}}{\sqrt{27}}\).

Step 4: Simplify Each Square Root

Simplify \(\sqrt{25}\) to 5, since 25 is a perfect square. Simplify \(\sqrt{27}\) to \(3\sqrt{3}\) because 27 is \(3^3\), and you can take out \(\sqrt{9} = 3\), leaving \(\sqrt{3}\).

Step 5: Combine the Results

Combine the simplified forms: \(\frac{5}{3\sqrt{3}}\).

Step 6: Rationalize the Denominator

Multiply both the numerator and the denominator by \(\sqrt{3}\) to remove the radical in the denominator. This gives \(\frac{5\sqrt{3}}{9}\).

Key Concepts

Prime Factorization

Prime factorization involves breaking down a number into its smallest prime factors. It’s a method used to simplify numbers by expressing them as a multiplication of prime numbers.
For instance, when simplifying \[\sqrt{\frac{75}{81}}\] in the original exercise, we first needed to simplify the fraction to \(\frac{25}{27}\). Prime factorization tells us that 25 is composed of two 5s, as \(25 = 5^2\), and 27 is made up of three 3s, or \(27 = 3^3\).

• This breakdown reveals that 25 and 27 are not perfect square ratios, so we must simplify further.
• Prime factorization is essential to identify the parts of a number we can simplify, particularly when dealing with square roots.

Understanding this helps us transform our initial expressions into simpler forms for easier mathematical manipulation.

Rationalizing the Denominator

Rationalizing the denominator involves removing the square root or the radical from the denominator of a fraction.
This is often required because a simpler form of a fraction without recourse to square roots in its denominator is more conventional in mathematics. In the original solution, after simplifying to \[\frac{5}{3\sqrt{3}}\] this step was necessary:

• We multiply both the numerator and the denominator by \(\sqrt{3}\), which is the radical present in the denominator, to maintain the equality of the expression.
• This yields \(\frac{5\sqrt{3}}{9}\), a form without radicals in the denominator.

The technique keeps equations tidy and adheres to standard formatting expectations, making them universally easier to comprehend and apply.

Square Roots Simplification

Simplifying square roots means reducing the expression under the radical sign to its simplest form.
This is achieved by factoring the expression and extracting the perfect squares. In our step-by-step solution:

• We simplified \(\sqrt{25}\) to 5, as 25 is a perfect square, leaving a clean expression free from radicals.
• The square root of 27 became \(3\sqrt{3}\), after identifying 9 (a perfect square) within 27, hence \(\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}\).

Recognizing perfect squares and simplifying them at an early stage during solving helps prevent errors and leads to faster solutions. It is a crucial skill for understanding complex mathematical problems and making calculations simpler and more efficient.