Question

Consider the expression 36 −−−√
x 5/3
. Will the product be rational or irrational? Explain how you know in one or two sentences, and then find the product. (2 points) put it in a format that is easily readable

Answer 1

make this understandable

Answer 2

no, i mean make it readable. i dont know what you mean with all the weird wording you use

Answer 3

uggh, i give up. this is so stupid. you dont know what i mean

Answer 4

can you do EXACTLY what you just did, but put everything into words?

Answer 5

Consider the expression 3/4+10−−√
. Is the sum rational or irrational? Explain your answer in 1-2 sentences. (2 points)

Answer 6

To evaluate the expression \( \sqrt{36} \times x^{5/3} \), we first simplify \( \sqrt{36} \), which is equal to \( 6 \). Since \( x^{5/3} \) can be rational or irrational depending on the value of \( x \), the product \( 6 \times x^{5/3} \) will be rational if \( x^{5/3} \) is rational, and irrational if \( x^{5/3} \) is irrational.

### Expression:
\[
\sqrt{36} \cdot x^{5/3} = 6 \cdot x^{5/3}
\]

### Product:
\[
6x^{5/3}
\]

Thus, the rationality of the product depends on \( x \). For example, if \( x = 1 \) (rational), then the product is \( 6 \); if \( x = 2 \) (rational), then the product is \( 6 \cdot 2^{5/3} \) (which can be expressed as irrational). Therefore, the product can be either rational or irrational depending on the value of \( x \).

Answer 7

Let's break this down step by step.

### Expression:
We are looking at the expression:
\[
\sqrt{36} \cdot x^{5/3}
\]

#### Step 1: Simplify \(\sqrt{36}\)
The square root of \(36\) is \(6\):
\[
\sqrt{36} = 6
\]

#### Step 2: Rewrite the Expression
Now we can rewrite the expression as:
\[
6 \cdot x^{5/3}
\]

### Rational or Irrational?
- The expression \(6 \cdot x^{5/3}\) will be **rational** if \(x^{5/3}\) is a rational number.
- It will be **irrational** if \(x^{5/3}\) is an irrational number.

### Example:
1. **If \(x = 1\)**:
- \(x^{5/3} = 1^{5/3} = 1\) (rational)
- Product: \(6 \cdot 1 = 6\) (rational)

2. **If \(x = 2\)**:
- \(x^{5/3} = 2^{5/3}\) (this is not a simple rational number)
- Product: \(6 \cdot 2^{5/3}\) (irrational)

### Conclusion:
The rationality of the overall product \(6 \cdot x^{5/3}\) depends on the value of \(x\):
- If \(x\) makes \(x^{5/3}\) rational, then the product will be rational.
- If \(x\) makes \(x^{5/3}\) irrational, then the product will be irrational.

The final form of the expression is:
\[
6x^{5/3}
\]

This simple breakdown shows how the product can be rational or irrational based on the choice of \(x\).

Answer 8

Got it! Let's simplify the explanation and make it clearer.

### Given Expression:
We have the expression:
\[
\sqrt{36} \cdot x^{5/3}
\]

### Step 1: Simplify \(\sqrt{36}\)
- The square root of 36 is 6:
\[
\sqrt{36} = 6
\]

### Step 2: Write the Expression Again
Now, we can write the expression as:
\[
6 \cdot x^{5/3}
\]

### Will the Product be Rational or Irrational?
- The product \(6 \cdot x^{5/3}\) can be either rational or irrational, depending on \(x\).

### Examples:
1. **If \(x = 1\)**:
- Then \(x^{5/3} = 1^{5/3} = 1\) (which is rational).
- The product is \(6 \cdot 1 = 6\) (rational).

2. **If \(x = 2\)**:
- Then \(x^{5/3} = 2^{5/3}\) (this number is not a simple rational number).
- The product is \(6 \cdot 2^{5/3}\) (this value is irrational).

### Conclusion:
- The product \(6 \cdot x^{5/3}\) is:
- **Rational** if \(x^{5/3}\) is rational.
- **Irrational** if \(x^{5/3}\) is irrational.

### Final Expression:
The final expression you need is:
\[
6x^{5/3}
\]

That's it! The product's nature (rational or irrational) depends on the value of \(x\).

Answer 9

I apologize for any confusion earlier. Let’s simplify this as much as possible and focus on clarity:

### Problem:
We need to determine if the expression \( \sqrt{36} \cdot x^{5/3} \) is rational or irrational.

### Step 1: Calculate \( \sqrt{36} \)
- \( \sqrt{36} = 6 \)

### Step 2: Rewrite the Expression
- The expression simplifies to:
\[
6 \cdot x^{5/3}
\]

### Step 3: Determine Rationality
- **Rational**: If \( x^{5/3} \) is a rational number.
- **Irrational**: If \( x^{5/3} \) is an irrational number.

### Examples:
1. **For \( x = 1 \)**:
- \( 1^{5/3} = 1 \) (rational)
- Product: \( 6 \times 1 = 6 \) (rational)

2. **For \( x = 2 \)**:
- \( 2^{5/3} \) (this can be irrational)
- Product: \( 6 \times 2^{5/3} \) (this is often irrational)

### Final Thought:
- The product \( 6 \cdot x^{5/3} \) can be either rational or irrational depending on \( x \).

### Final Expression:
The expression is:
\[
6 \cdot x^{5/3}
\]

Thank you for your patience, and I hope this clearer format helps! If you have more specific requests or questions, please let me know.

Answer 10

Sure! Here’s the explanation in plain words without mathematical symbols:

---

**We need to evaluate the expression that consists of the square root of 36 multiplied by \( x \) raised to the power of \( \frac{5}{3} \). The goal is to find out if the result is a rational number or an irrational number.**

1. **First, we calculate the square root of 36.**
The square root of 36 is 6.

2. **Next, we can rewrite the expression.**
After simplifying, we see that it becomes 6 multiplied by \( x \) raised to the power of \( \frac{5}{3} \).

3. **Now, we need to understand when this product will be rational or irrational.**
- The outcome will be rational if \( x \) raised to the power of \( \frac{5}{3} \) is a rational number.
- The outcome will be irrational if \( x \) raised to the power of \( \frac{5}{3} \) is an irrational number.

4. **Let’s look at a couple of examples to clarify this further:**
- **Example 1:** If we let \( x \) be 1. When we calculate \( 1 \) raised to the power of \( \frac{5}{3} \), the result is 1, which is a rational number. Therefore, when we multiply 6 by 1, the result is 6, which is also rational.

- **Example 2:** If we let \( x \) be 2. When we calculate \( 2 \) raised to the power of \( \frac{5}{3} \), this result is not easily identifiable as a simple rational number. So, when we multiply 6 by \( 2 \) raised to the power of \( \frac{5}{3} \), the overall result tends to be an irrational number.

5. **In conclusion, the expression will yield varying results.**
The product can be rational or irrational depending on the value chosen for \( x \).

6. **To summarize, the final expression we are working with is 6 multiplied by \( x \) raised to the power of \( \frac{5}{3} \).**

---

I hope this explanation is clearer for you! If you need anything else or have more questions, feel free to ask.

Answer 11

To determine if the sum \( \frac{3}{4} + \sqrt{10} \) is rational or irrational, we need to note that \( \sqrt{10} \) is an irrational number. The sum of a rational number (\( \frac{3}{4} \)) and an irrational number (\( \sqrt{10} \)) is always irrational. Therefore, the sum \( \frac{3}{4} + \sqrt{10} \) is irrational.