Problem 27 - Algebra 1

Question

Simplify the expression. $$3 \sqrt{\frac{9}{3}}$$

Step-by-Step Solution

Answer

The simplified answer is \(3\sqrt{3}\)

Step 1: Simplify Fraction

First simplify the fraction inside the square root. \( \frac{9}{3} = 3 \)

Step 2: Solve Square Root

Next, compute the square root of 3. \( \sqrt{3} = \sqrt{3} \)

Step 3: Multiply Result

Lastly, multiply the coefficient (3) by the result from taking the square root of 3. The answer is \(3 * \sqrt{3} = 3\sqrt{3}\)

Key Concepts

Radical Expressions

Thinking about radical expressions involves working with the likes of square roots, cube roots, and higher-degree roots. These expressions include a radical symbol \( \sqrt{} \) to represent the root operation, and can appear quite daunting for first-time learners. A key goal when dealing with radical expressions is to simplify them to their most basic form. This often means looking inside the radical to see if there are any factors that are perfect squares (in the case of square roots) or perfect cubes (for cube roots), and then simplifying accordingly.

For instance, in our example \(3 \sqrt{\frac{9}{3}}\), we must first address the fraction within the radical. Why? Because simplifying anything inside a radical can potentially make the next steps easier. Plus, it is a standard mathematical procedure to simplify an expression before performing other operations on it. And that's exactly what we do by converting \(\frac{9}{3}\) to 3 before we proceed with further simplification.

Simplifying radical expressions is just a matter of breaking down complex-looking numbers into simpler components. The process involves skills that are fundamental in mathematics, such as factorization and understanding the properties of numbers. As such, learning to manipulate these expressions can enhance your overall algebra proficiency.

Square Roots

A square root essentially asks, 'What number, when multiplied by itself, gives me this number under the square root symbol?' It’s a fundamental concept in algebra and appears frequently in various mathematical contexts. When you see a symbol like \( \sqrt{ }\), it denotes the principal square root of the number inside.

The square root of a perfect square, such as \( \sqrt{9} \) which equals 3, is straightforward. However, not all numbers are perfect squares, which means their square roots can be more complex, often irrational numbers (numbers that can't be expressed as simple fractions), like \( \sqrt{2} \) or \( \sqrt{3} \).

In our original problem, \( \sqrt{\frac{9}{3}} \) becomes \( \sqrt{3} \) after simplifying the fraction. \( \sqrt{3} \) is not a perfect square, so it remains 'as is' unless it's part of a larger equation where it may be further simplified. Understanding square roots is not just about finding the root; it’s also about recognizing when a number is in its simplest radical form, and that sometimes, that form doesn't look 'neat' or 'whole'.

Simplifying Fractions

Simplifying fractions is often one of the first steps in solving problems involving algebra and geometry. It entails reducing the fraction to its simplest form, where the numerator and the denominator are as small as possible and have no common factors other than 1. This is accomplished by dividing both the numerator and the denominator by their greatest common divisor (GCD).

With the fraction \( \frac{9}{3} \), it's easy to see that the GCD is 3. When we divide both 9 and 3 by this number, we're left with 1 in the denominator and 3 in the numerator, simplifying the fraction to 3. This might seem trivial with small numbers, but the principle is crucial when dealing with more complex equations and larger numbers.

Understanding how to simplify fractions allows students to tackle problems methodically, leading to cleaner and often easier to understand solutions. Whether it's part of a radical expression or a polynomial division, mastering this skill is invaluable. It's a stepping stone that plays a significant role in the comprehension and implementation of various mathematical operations.