https://file.aiquickdraw.com/mathbot/file/1725367884158apcrd3zn.jpg

Let's analyze and solve the three questions in the image.

### 1. **First Question:**
   \[
   \text{Solve } \frac{2x^3}{x} = 5x^7
   \]

#### Solution:
Start by simplifying the left side of the equation:
\[
\frac{2x^3}{x} = 2x^{3-1} = 2x^2
\]

So, the equation becomes:
\[
2x^2 = 5x^7
\]

Next, we can divide both sides by \( x^2 \) (assuming \( x \neq 0 \)):
\[
2 = 5x^{7-2} = 5x^5
\]

Now, solve for \( x \):
\[
x^5 = \frac{2}{5}
\]
\[
x = \left(\frac{2}{5}\right)^{\frac{1}{5}}
\]

So, \( x = \left(\frac{2}{5}\right)^{\frac{1}{5}} \).

### 2. **Second Question:**
   \[
   \text{Find the cube root of 1331 and 2197. Are these rational or irrational?}
   \]

#### Solution:
- The cube root of 1331 is calculated as follows:
  \[
  \sqrt[3]{1331} = 11
  \]
  Since 11 is a rational number, the cube root of 1331 is rational.

- The cube root of 2197 is calculated as follows:
  \[
  \sqrt[3]{2197} = 13
  \]
  Since 13 is a rational number, the cube root of 2197 is rational.

Therefore, both cube roots are rational numbers.

### 3. **Third Question:**
   \[
   \text{If the perimeter of the triangle ABC is equal to } x + y + z, \text{find the length of AB if AC = } y, \text{BC = } z, \text{and AB = } x.
   \]

#### Solution:
Given that the perimeter of triangle ABC is equal to \( x + y + z \), and the lengths of the sides are given by:
- \( AB = x \)
- \( AC = y \)
- \( BC = z \)

The problem is straightforward since it asks for the length of AB:
\[
\text{AB} = x
\]

Since the perimeter is simply the sum of all the sides:
\[
\text{Perimeter} = AB + AC + BC = x + y + z
\]

Thus, \( AB = x \) as given.

---

Do you have any questions or need further details?

### 5 Related Questions:
1. How can we determine if a root is rational or irrational?
2. What are the general rules for simplifying equations involving exponents?
3. How would the solution to the first problem change if \( x = 0 \) is allowed?
4. How can cube roots of non-perfect cubes be approximated?
5. What is the geometric significance of knowing the perimeter in a triangle?

### Tip:
When solving equations involving exponents, always check for potential solutions where the variable could equal zero, as this might offer additional valid solutions.

Learn how to solve equations involving exponents, find cube roots, and calculate triangle perimeter. This detailed solution covers key concepts like exponent rules and cube root calculations. Suitable for students in Grades 7-9.

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