Problem 1 What does it mean to rationalize... [FREE SOLUTION]

Chapter 10: Problem 1

What does it mean to rationalize the denominator of a radical expression?

Short Answer

Expert verified

To rationalize the denominator of a radical expression means to eliminate radical expressions from the denominator of a fraction. This is typically done by multiplying both the numerator and denominator by a suitable value, such as the conjugate, to eliminate the radical without changing the expression's value. For example, given \(\frac{1}{\sqrt{2}}\), we multiply it by \(\frac{\sqrt{2}}{\sqrt{2}}\) to obtain the simplified, rationalized expression \(\frac{\sqrt{2}}{2}\).

Step by step solution

To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a number is the same number, but with the opposite sign in its middle term. In this case, the conjugate of \(\sqrt{2}\) is \(\sqrt{2}\) itself, as there is no middle term. #Step 2: Multiply by the Conjugate#

We will multiply the expression by \(\frac{\sqrt{2}}{\sqrt{2}}\), as this will allow us to eliminate the radical from the denominator: \(\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}\) #Step 3: Simplify the Numerator#

Now, we multiply the numerators: \(\frac{1 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}}\) This gives us: \(\frac{\sqrt{2}}{2}\) #Step 4: Check the Simplified Expression#

The simplified expression is \(\frac{\sqrt{2}}{2}\). The denominator has been successfully rationalized, as there are no more radicals in the denominator. In some instances, further simplification may be required, but in our example, this is the final answer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Radical Expressions

Radical expressions are mathematical expressions that include a radical sign \( \sqrt{} \). This sign indicates the root of a number. The most common radical is the square root, but we can also have cube roots, fourth roots, and so on.
Radicals are quite useful because they let us work with irrational numbers, numbers that cannot be represented as a simple fraction. When dealing with radical expressions, it's crucial to know how to manipulate them for simplification and easier calculation.
When we have a radical expression as a denominator, it can complicate further calculations or comparisons with numbers, which is why we often aim to rationalize the denominator.

The Role of Conjugates in Rationalization

Conjugates are essential tools in the process of rationalizing denominators. The conjugate of a binomial expression is similar to the original expression but with a reversed sign in the middle.
For example, if our expression is \( a + b \), its conjugate is \( a - b \). This concept is pivotal when dealing with denominators containing radicals.

Applying the conjugate effectively removes the radical after multiplying, thanks to the formula \( (a + b)(a - b) = a^2 - b^2 \).
Using this product results in a rational number, i.e., a number with no radicals in the denominator.

In our case involving a simple radical such as \( \sqrt{2} \), the conjugate is \( \sqrt{2} \) itself, since it's the simplest form with no middle term involved.

The Process of Simplification

Simplification is the step of making an expression more manageable and easier to work with. After multiplying by the conjugate, you often end up with both radicals and non-radical parts.
The goal is to combine like terms and simplify the expression as much as possible.

After multiplying the numerator and the denominator by the conjugate, you may find that clustering similar terms helps.
This includes rewriting fractions, reducing terms, and ensuring no radical remains in places like the denominator.

In our exercise, multiplying \( \frac{1}{\sqrt{2}} \) by the conjugate \( \frac{\sqrt{2}}{\sqrt{2}} \) transformed the expression into \( \frac{\sqrt{2}}{2} \), a rational form. This new form is easier to interpret and use in calculations.

Rationalizing the Denominator

The main goal of rationalizing the denominator is to remove any radicals present in the denominator. A denominator without radicals is often more convenient for further mathematical manipulation and interpretation.
Here's how it's generally done:

Identify the radical or radicals in the denominator.
Multiply both the numerator and denominator by the conjugate of the denominator.
Perform the multiplication to leave a rational number.

For instance, when starting with \( \frac{1}{\sqrt{2}} \) and multiplying by \( \frac{\sqrt{2}}{\sqrt{2}} \), the radicals in the denominator get eliminated. The result, \( \frac{\sqrt{2}}{2} \), clearly has no radicals in the denominator, achieving the objective of rationalization.

91影视

What does it mean to rationalize the denominator of a radical expression?

Short Answer

Step by step solution

Now, we multiply the numerators: \(\frac{1 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}}\) This gives us: \(\frac{\sqrt{2}}{2}\) #Step 4: Check the Simplified Expression#

Key Concepts

Understanding Radical Expressions

The Role of Conjugates in Rationalization

The Process of Simplification

Rationalizing the Denominator

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