Problem 3 - Algebra 2

Question

Simplify. $$ \sqrt{\frac{48}{49}} $$

Step-by-Step Solution

Answer

\( \frac{4\sqrt{3}}{7} \)

Step 1: Understand the Expression

The expression to simplify is \( \sqrt{\frac{48}{49}} \). This is a square root of a fraction, where the numerator is 48 and the denominator is 49.

Step 2: Find the Square Roots of the Numerator and Denominator

The square root of a fraction can be simplified by finding the square root of the numerator and the square root of the denominator separately. This means we need to compute \( \sqrt{48} \) and \( \sqrt{49} \).

Step 3: Simplify the Denominator

The square root of the denominator is \( \sqrt{49} = 7 \). This is because 7 multiplied by itself gives 49.

Step 4: Simplify the Numerator

48 can be factored into \( 16 \times 3 \). Since 16 is a perfect square, \( \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \).

Step 5: Write the Simplified Expression

Now that we have simplified both the numerator and the denominator, we can write the entire expression as:\[\frac{4\sqrt{3}}{7}\]

Key Concepts

Square roots of fractions

Dealing with the square root of a fraction can seem tricky, but it's quite straightforward. The key idea is to break it down into simpler parts. When you have an expression like \( \sqrt{\frac{48}{49}} \), you can simplify it by finding the square root of both the numerator and the denominator separately.

This method leverages the property \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). By applying this, you get a simpler fraction that is often easier to work with or further simplify. This process involves handling each component—numerator and denominator—one at a time, leading to a simplified final expression, as you see with \( \frac{4\sqrt{3}}{7} \). Keep this property in mind, and you'll find simplifying square roots in fraction form much more manageable.

Perfect squares

Perfect squares play a vital role when simplifying square root expressions, and knowing what they are can make your life much easier. A perfect square is a number that results from an integer multiplied by itself.

For example:

• \(4\) is a perfect square because \(2 \times 2 = 4\)
• \(9\) is a perfect square because \(3 \times 3 = 9\)
• \(16\) is a perfect square because \(4 \times 4 = 16\)

Recognizing perfect squares allows you to quickly identify and simplify square roots in expressions. In the case of \( \sqrt{48} \), you notice that \(48\) can be factored into \(16 \times 3\), where \(16\) is a perfect square. This recognition is pivotal in simplifying \( \sqrt{48} \) into \(4\sqrt{3}\). Understanding perfect squares will boost your efficiency in simplifying expressions involving roots.

Numerator and denominator

In fractions, we see two main parts: the numerator and the denominator. The numerator is the top number, and the denominator is the bottom one. These parts work together to form a fraction.

Understanding these elements helps in applying operations like simplification. In the problem \(\sqrt{\frac{48}{49}}\), 48 is the numerator, and 49 is the denominator. By addressing each part's square root separately, you can simplify the fraction effectively.

• The numerator \(\sqrt{48}\) was simplified by breaking it into a product of factors as \(4\sqrt{3}\).
• The denominator \(\sqrt{49}\) is simply \(7\) since 49 is a perfect square.

This step-by-step breakdown of dealing with numerators and denominators in fractions is crucial for proper simplification and is essential for building a strong mathematical foundation.

Factoring numbers

Factoring numbers is a powerful skill, especially when simplifying square roots. To factor a number means to break it down into smaller numbers or 'factors' that multiply to make the original number.

In the context of simplifying \(\sqrt{48}\), the number 48 was broken down into \(16 \times 3\). Recognizing that 16 is a perfect square helps simplify \( \sqrt{48} \) to \(4\sqrt{3}\).

• Identify numbers that multiply to give you the number you're factoring.
• Look for perfect squares among those factors because they simplify easily.
• Use these perfect squares to simplify your square roots effectively.

This method enables you to break down complex expressions into simpler, more manageable forms. With practice, factoring becomes an intuitive skill that greatly enhances your mathematical efficiency.