In each of the following questions, two equations are given. You have to solve them and give answer(निम्नलिखित प्रत्येक प्रश्न में दो समीकरण दिए गए हैं। आपको उन्हें हल करना है और उत्तर देना है
I. `2x^(5)(x^(-2))=128`
II. `1/(3)y^(9)=1/24 y^(11)`

Options

A. `xgty`
B. `xgey`
C. `x lty`
D. `xley`

Answer

To solve the given equations step by step, let's start with the first equation:

### Step 1: Solve the first equation
The first equation is:
\[ 2x^{5} \cdot x^{-2} = 128 \]

Using the laws of exponents, we can combine the terms with the same base:
\[ 2x^{5 + (-2)} = 128 \]
This simplifies to:
\[ 2x^{3} = 128 \]

### Step 2: Isolate \( x^{3} \)
Next, we will isolate \( x^{3} \) by dividing both sides by 2:
\[ x^{3} = \frac{128}{2} \]
This simplifies to:
\[ x^{3} = 64 \]

### Step 3: Find \( x \)
Now, we take the cube root of both sides to find \( x \):
\[ x = \sqrt[3]{64} \]

Since \( 64 = 4^3 \), we have:
\[ x = 4 \]

### Step 4: Solve the second equation
The second equation is:
\[ \frac{1}{3}y^{9} = \frac{1}{24}y^{11} \]

### Step 5: Clear the fractions
To eliminate the fractions, we can multiply both sides by 24:
\[ 24 \cdot \frac{1}{3}y^{9} = y^{11} \]
This simplifies to:
\[ 8y^{9} = y^{11} \]

### Step 6: Rearrange the equation
Now, we can rearrange the equation:
\[ y^{11} - 8y^{9} = 0 \]
Factoring out \( y^{9} \):
\[ y^{9}(y^{2} - 8) = 0 \]

### Step 7: Solve for \( y \)
Setting each factor to zero gives us:
1. \( y^{9} = 0 \) (which implies \( y = 0 \))
2. \( y^{2} - 8 = 0 \) (which implies \( y^{2} = 8 \))

Taking the square root of both sides gives:
\[ y = \sqrt{8} = 2\sqrt{2} \quad \text{or} \quad y = -\sqrt{8} = -2\sqrt{2} \]

### Step 8: Compare \( x \) and \( y \)
Now we have:
- \( x = 4 \)
- \( y = 2\sqrt{2} \) or \( y = -2\sqrt{2} \)

Since \( 2\sqrt{2} \) is approximately \( 2.83 \), we can see that:
- \( 4 > 2\sqrt{2} \)
- \( 4 > -2\sqrt{2} \)

Thus, we conclude that:
\[ x > y \]

### Final Answer
The answer is \( x > y \).

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