https://file.aiquickdraw.com/mathbot/file/1725416458891qay2ax4z.jpg

To solve the problem shown in the image, let's find the product of \( 4\sqrt{7} \) and \( 5\sqrt{3} \), simplify it, and determine whether the result is rational or irrational.

### Step 1: Multiply the two expressions
\[
4\sqrt{7} \times 5\sqrt{3}
\]
Multiply the coefficients (4 and 5) and the square roots (\(\sqrt{7}\) and \(\sqrt{3}\)) separately:
\[
4 \times 5 = 20
\]
\[
\sqrt{7} \times \sqrt{3} = \sqrt{7 \times 3} = \sqrt{21}
\]
So the product is:
\[
20\sqrt{21}
\]

### Step 2: Simplify the expression
The expression \( 20\sqrt{21} \) is already in its simplest form because \( \sqrt{21} \) cannot be simplified further.

### Step 3: Determine if the result is rational or irrational
A number is rational if it can be expressed as the ratio of two integers and has a decimal expansion that terminates or repeats. Here, \( 20\sqrt{21} \) contains \( \sqrt{21} \), which is an irrational number (since 21 is not a perfect square). Therefore, \( 20\sqrt{21} \) is also irrational.

### Final Answer:
- **Result:** \( 20\sqrt{21} \)
- **Explanation:** The result is **irrational** because it contains \( \sqrt{21} \), which cannot be written as the ratio of two integers and its decimal expansion neither terminates nor repeats.

Would you like more details or have any questions about this solution?

Here are five additional questions related to this topic:
1. How would the product change if one of the square roots was a rational number?
2. What is the product of \( 4\sqrt{7} \) and \( \sqrt{7} \)?
3. How can you prove that \( \sqrt{21} \) is irrational?
4. What is the general rule for multiplying square roots?
5. How would you express \( 20\sqrt{21} \) in decimal form approximately?

**Tip:** When multiplying square roots, always simplify the square root part before multiplying if possible.

Find the product of 4√7 and 5√3 in simplest form. Also, determine whether the result is rational or irrational and explain your answer.

Learn how to find the product of 4√7 and 5√3, simplify the expression, and determine whether the result is rational or irrational. This problem covers key algebraic concepts including the multiplication of radicals and the distinction between rational and irrational numbers. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation