https://r2.erweima.ai/mathbot/image/1726061704858hmsy6vnh.jpg

To rationalize the denominator of the expression given in part (b):

\[
\frac{3c - \sqrt{c}}{\sqrt{c}}
\]

we need to eliminate the square root from the denominator. Here are the steps:

### Step 1: Multiply the numerator and the denominator by \(\sqrt{c}\)

\[
\frac{3c - \sqrt{c}}{\sqrt{c}} \times \frac{\sqrt{c}}{\sqrt{c}} = \frac{(3c - \sqrt{c}) \cdot \sqrt{c}}{c}
\]

### Step 2: Distribute \(\sqrt{c}\) in the numerator

\[
\frac{3c \cdot \sqrt{c} - (\sqrt{c})^2}{c}
\]

Simplify the powers:

\[
\frac{3c \sqrt{c} - c}{c}
\]

### Step 3: Split the fraction into two parts

\[
\frac{3c \sqrt{c}}{c} - \frac{c}{c}
\]

### Step 4: Simplify each term

1. \(\frac{3c \sqrt{c}}{c} = 3 \sqrt{c}\)
2. \(\frac{c}{c} = 1\)

Thus, the expression simplifies to:

\[
3 \sqrt{c} - 1
\]

### Final Answer

The rationalized form of the expression is:

\[
3 \sqrt{c} - 1
\]

Would you like further details on this solution or have any questions? 

Here are 5 related questions that could expand your understanding:

1. What is the process of rationalizing the denominator and why is it used?
2. How do you rationalize a denominator when it has a binomial form, such as \((a + \sqrt{b})\)?
3. What happens if both the numerator and denominator contain square roots?
4. How can you simplify expressions involving higher roots, like cube roots?
5. How would the process change if \(c\) were not a prime number?

**Tip:** When rationalizing denominators, multiplying by the conjugate (for expressions with sums or differences involving roots) can simplify more complex expressions.

Given that c is a prime number, rationalize the denominator of (3c - √c)/√c. Simplify your answer.

Learn how to rationalize the denominator of the expression (3c - √c)/√c where c is a prime number. This problem involves key algebraic concepts such as rationalization, simplification, and properties of radicals. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation