Problem 79 - Algebra 2

Question

Simplify each expression. \(\frac{\sqrt{5}}{\sqrt{8}}\)

Step-by-Step Solution

Answer

\( \frac{\sqrt{10}}{4} \)

Step 1: Recognize the problem

We need to simplify the expression \( \frac{\sqrt{5}}{\sqrt{8}} \). This expression involves dividing one square root by another, which is not in its simplest form.

Step 2: Use the property of square roots

Apply the property of square roots that states \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). Therefore, \( \frac{\sqrt{5}}{\sqrt{8}} = \sqrt{\frac{5}{8}} \).

Step 3: Simplify the fraction inside the square root

Notice that \( \frac{5}{8} \) is already in its simplest form, so no further simplification is needed inside the square root.

Step 4: Rationalize the denominator

To simplify \( \sqrt{\frac{5}{8}} \) and ensure the result does not have a square root in the denominator, we can multiply both numerator and denominator by \( \sqrt{8} \): \( \frac{\sqrt{5} \cdot \sqrt{8}}{\sqrt{8} \cdot \sqrt{8}} = \frac{\sqrt{40}}{8} \).

Step 5: Simplify the numerator

Simplify \( \sqrt{40} \) as \( \sqrt{4 \cdot 10} = 2 \sqrt{10} \). This gives \( \frac{2 \sqrt{10}}{8} \).

Step 6: Final simplification

Reduce \( \frac{2 \sqrt{10}}{8} \) by dividing numerator and denominator by 2: \( \frac{2 \sqrt{10}}{8} = \frac{\sqrt{10}}{4} \). This is the simplest form.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is a crucial step in simplifying expressions involving square roots. The goal here is to eliminate any square roots from the denominator. This practice makes expressions simpler to work with and understand.

To rationalize a denominator, you multiply both the numerator and denominator by a value that will cancel out the square root in the denominator. For instance, in the expression \( \frac{\sqrt{5}}{\sqrt{8}} \), the denominator is \( \sqrt{8} \).

• Multiply both the numerator and denominator by \( \sqrt{8} \).
• This process doesn’t change the value of the expression because you're essentially multiplying by 1, or \( \frac{\sqrt{8}}{\sqrt{8}} \).

After this multiplication, you are left with \( \frac{\sqrt{40}}{8} \). The denominator is now rational, meaning it is a regular number without a square root.

Properties of Square Roots

Understanding the properties of square roots helps simplify expressions efficiently. One fundamental property is that \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). This allows us to combine the square roots into a single square root of a fraction.

Using this property simplifies expressions like \( \frac{\sqrt{5}}{\sqrt{8}} \) to \( \sqrt{\frac{5}{8}} \). Yet, further steps are often needed:

• You can transform the problem into a single radius (i.e., \( \sqrt{\frac{5}{8}} \)).
• Simplifying fractions within a square root is sometimes impossible if the fraction is already in simplest terms.

Bear in mind that the property helps reduce complexities when faced with larger or more cumbersome numbers under the root sign.

Fraction Simplification

Fraction simplification is about reducing fractions to their simplest form. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1.

In our example, after using properties of square roots and rationalizing the denominator, we had \( \frac{2\sqrt{10}}{8} \). To simplify this fraction, divide both the numerator and denominator by their greatest common factor. Here:

• The GCF of 2 (from the numerator) and 8 (from the denominator) is 2.
• Dividing both parts by 2 gives \( \frac{\sqrt{10}}{4} \).

The final result is the simplest form of the expression with a rationalized denominator and simplified fraction, illustrating a clean and understandable result.