Problem 3 - Intermediate Algebra

Question

To ____ the denominator of \(\frac{4}{\sqrt{5}},\) we multiply the fraction by \(\frac{\sqrt{5}}{\sqrt{5}}\)

Step-by-Step Solution

Answer

To rationalize the denominator, multiply by \( \frac{\sqrt{5}}{\sqrt{5}} \). The simplified form is \( \frac{4\sqrt{5}}{5} \).

Step 1: Understanding the Question

The exercise asks us to simplify the fraction \( \frac{4}{\sqrt{5}} \). Simplifying by removing the square root from the denominator is achieved by a process known as rationalizing the denominator.

Step 2: Identifying the Method

To rationalize the denominator of \( \frac{4}{\sqrt{5}} \), we need to eliminate \( \sqrt{5} \) from the denominator by multiplying both the numerator and the denominator by \( \sqrt{5} \).

Step 3: Setting Up the Expression

Multiply the fraction \( \frac{4}{\sqrt{5}} \) by \( \frac{\sqrt{5}}{\sqrt{5}} \), resulting in the expression \( \frac{4 \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} \).

Step 4: Simplifying the Denominator

\( \sqrt{5} \cdot \sqrt{5} \) simplifies to \( 5 \), since \( \sqrt{a} \cdot \sqrt{a} = a \), for any non-negative real number \( a \).

Step 5: Simplifying the Numerator

Multiply the numerators together: \( 4 \cdot \sqrt{5} = 4\sqrt{5} \).

Step 6: Combining and Simplifying the Fraction

The resulting expression is \( \frac{4\sqrt{5}}{5} \), which is the simplified form of the original fraction with a rationalized denominator.

Key Concepts

Simplifying Fractions

Simplifying fractions means rewriting a fraction so that the numerator and the denominator have no common factors other than one. It is like cleaning up the fraction to its simplest form. But in some cases, simplifying goes beyond just finding common factors and involves making the expression more straightforward, such as when dealing with square roots in the denominator.
To simplify an expression like this one, the process involves rationalizing the denominator, which means eliminating square roots from the bottom of the fraction. This might seem a bit tricky, but it is necessary to ensure that fractions remain in a standard, clean format. This method ensures that calculations with these fractions, like addition or comparison, become much easier and more straightforward.

• For rationalizing, look for square roots or complex numbers in the denominator.
• Multiply by a form of one that helps eliminate the square root or imaginary part.
• Ensure that the fraction is now in its simplest form by checking for common factors.

Rationalizing and simplifying thus go hand in hand, making sure that the fraction is in a much friendlier form for further calculations.

Square Roots

Square roots are mathematical operations that help us determine what number, when multiplied by itself, gives a certain number. Understanding square roots is crucial when working with fractions, especially when they appear in the denominator.
In the example of simplifying the fraction \( \frac{4}{\sqrt{5}} \), we encounter the square root of 5. If left unchecked, this can make arithmetic cumbersome due to irrational numbers popping up in the denominator. This is precisely why it's important to rationalize the fraction by removing the square root from the bottom.
When you multiply a square root by itself, you are left with the original number beneath the root. For example, \( \sqrt{5} \cdot \sqrt{5} = 5 \). This property is what allows us to transform the fraction into something more manageable. The end goal with square roots in denominators is to get rid of them, hence simplifying the arithmetic handling of the expression.

Numerator and Denominator

The terms numerator and denominator are fundamental parts of a fraction. The numerator is the top part of the fraction and represents how many parts we are considering. The denominator is the bottom part which states the total number of equal parts something is divided into.
In the context of rationalizing a denominator, like in \( \frac{4}{\sqrt{5}} \), these components play distinct roles. The main focus is initially on the denominator - here it contains the square root \( \sqrt{5} \), which we need to eliminate to simplify our calculations.
To achieve a simplified fraction, multiply both the numerator and denominator by the number that will help remove the square root from the bottom. In this case, multiplying by \( \sqrt{5} \) helps in getting a denominator without a square root, now appearing as 5 on the bottom, while the numerator becomes \( 4\sqrt{5} \).
This transformation allows the fraction to stay balanced while achieving a neat and easily manageable expression. It's all about making sure the fraction reflects the simplest, yet accurate, form of the values represented.