The ubiquitous AM-GM inequality has many
applications. It almost crops up in unlikely situations and
the solutions using AM-GM are truly elegant . Recall
that for n positive reals `a_(i) I = 1,2 …,`n, the AM-GM inequality tells
`(overset(n) underset(1)suma_i)/n ge ( overset(n)underset(1)proda_i)^((1)/(n))`
The special in which the inequality turns into equality
help solves many problems where at first we seem to
have not informantion to arrive at the answer .
The number of ordered pairs (x,y) pf real numbers
satisfying the equation `x^(8) + 6= 8 |xy|-y^(8)` is

Options

A. (a)10
B. (b)8
C. (c)4
D. (d)2

Answer

To solve the equation \( x^8 + 6 = 8 |xy| - y^8 \) and find the number of ordered pairs \((x, y)\) of real numbers that satisfy it, we can follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ x^8 + 6 = 8 |xy| - y^8 \]
Rearranging gives:
\[ x^8 + y^8 + 6 = 8 |xy| \]

### Step 2: Applying AM-GM Inequality
We can apply the AM-GM inequality to the terms \(x^8\), \(y^8\), and six instances of 1 (since \(6 = 1 + 1 + 1 + 1 + 1 + 1\)):
\[ \frac{x^8 + y^8 + 1 + 1 + 1 + 1 + 1 + 1}{8} \geq \sqrt[8]{x^8 \cdot y^8 \cdot 1^6} \]
This simplifies to:
\[ \frac{x^8 + y^8 + 6}{8} \geq |xy| \]

### Step 3: Equating AM and GM
From our rearranged equation, we have:
\[ \frac{x^8 + y^8 + 6}{8} = |xy| \]
By the AM-GM inequality, we know:
\[ |xy| \leq \frac{x^8 + y^8 + 6}{8} \]
Thus, we can conclude:
\[ |xy| = \frac{x^8 + y^8 + 6}{8} \]

### Step 4: Finding Conditions for Equality
The equality condition in AM-GM holds when all terms are equal. Therefore, we set:
\[ x^8 = y^8 = 1 \]
This gives us:
\[ x = 1 \text{ or } -1, \quad y = 1 \text{ or } -1 \]

### Step 5: Listing Ordered Pairs
The possible combinations of \((x, y)\) are:
1. \( (1, 1) \)
2. \( (1, -1) \)
3. \( (-1, 1) \)
4. \( (-1, -1) \)

Thus, we have a total of **4 ordered pairs**.

### Conclusion
The number of ordered pairs \((x, y)\) of real numbers satisfying the equation is:
\[ \boxed{4} \]