Problem 76 - Algebra for College Students

Question

Your friend simplified \(\frac{\sqrt{6}}{\sqrt{8}}\) as follows: $$ \frac{\sqrt{6}}{\sqrt{8}} \cdot \frac{\sqrt{8}}{\sqrt{8}}=\frac{\sqrt{48}}{8}=\frac{\sqrt{16} \sqrt{3}}{8}=\frac{4 \sqrt{3}}{8}=\frac{\sqrt{3}}{2} $$

Step-by-Step Solution

Answer

The simplified form of \( \frac{\sqrt{6}}{\sqrt{8}} \) is \( \frac{\sqrt{3}}{2} \).

Step 1: Recognize the Structure

We start with the initial expression \( \frac{\sqrt{6}}{\sqrt{8}} \). Notice that both the numerator and the denominator are square roots.

Step 2: Rationalize the Denominator

To rationalize the denominator, multiply the expression by \( \frac{\sqrt{8}}{\sqrt{8}} \). This utilizes the property of a "conjugate" to eliminate the square root in the denominator.

Step 3: Simplify the Numerator

Now simplify the numerator: \( \sqrt{6} \times \sqrt{8} = \sqrt{48} \). This step uses the property \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \).

Step 4: Simplify \(\sqrt{48}\)

Recognize that \( \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3} \). This rearranges the product inside the square root to separate perfect squares.

Step 5: Simplify the Expression

Now, rewrite the expression as \( \frac{4\sqrt{3}}{8} \). Simplify this fraction by canceling out the common factor 4, to get \( \frac{\sqrt{3}}{2} \).

Key Concepts

Simplifying Radical Expressions

Simplifying radical expressions involves rewriting an expression to its simplest form. Let's consider a radical expression like \( rac{\sqrt{6}}{\sqrt{8}} \).

This expression includes square roots in both the numerator and the denominator. Our goal is to simplify it to make it more manageable and easy to understand. This process usually involves removing the radical from the denominator so the expression is more visually appealing and easier to work with.

• Identify the radicals: Notice that \( \sqrt{6} \) and \( \sqrt{8} \) are radical expressions.
• Multiply appropriately: Use the concept of rationalizing the denominator by finding a suitable number to multiply both the numerator and the denominator. Here, multiply by \( \frac{\sqrt{8}}{\sqrt{8}} \).

The top and the bottom of the fraction are multiplied by the same non-zero value to ensure that the fraction's value remains unchanged, while allowing us to simplify the radicals.

Square Roots

Square roots are a type of mathematical expression where a number represents a value that, when multiplied by itself, gives the original number inside the radical.

Any number can be expressed as a square root. For example, \( \sqrt{8} \) is an expression that signifies the number when squared will give 8. You can break down a square root into factors:

• Find perfect squares: Check if there is a perfect square that is a factor of the number inside the square root.
• Simplify the expression: For example, \( \sqrt{48} \) can be rewritten as \( \sqrt{16 \times 3} \), which further breaks down to \( \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3} \).

This helps in simplifying complex expressions and making them easier to understand and work with.

Fraction Simplification

Simplifying fractions is a method to reduce a fraction to its simplest form. This means making the fraction as easy to work with as possible, which typically involves factoring out common values in the numerator and the denominator.

After rationalizing the expression \( \frac{4\sqrt{3}}{8} \), simplifying this fraction involves identifying and canceling common factors:

• Look for common factors: Here, both 4 and 8 can be divided by the same number, i.e., 4.
• Shrink the fraction: Divide both the numerator and the denominator by their greatest common factor. \( \frac{4\sqrt{3}}{8} \) simplifies to \( \frac{\sqrt{3}}{2} \).

This process reduces the expression to its most basic form, ensuring it's as streamlined as possible for further mathematical operations or interpretations.