If `F = a.(|b|).c.(|d|).e.(|f|).g(|h|).i.(|j|)` and `G = a.b.c. d.e.f.g.h.i.j` Again, `a = (-1)^1, b = (2)^(-2) , c = (-3)^(3), d = (4)^(-4) ,…..j = (10)^(-10)` then the correct relation is :

Options

A. `F =-G`
B. `F = G`
C. `F.G = 1`
D. None of these

Answer

To solve the problem, we need to evaluate the expressions for \( F \) and \( G \) given the values of \( a \) through \( j \).

### Step 1: Define the values of \( a \) through \( j \)

Given:
- \( a = (-1)^1 = -1 \)
- \( b = 2^{-2} = \frac{1}{2^2} = \frac{1}{4} \)
- \( c = (-3)^3 = -27 \)
- \( d = 4^{-4} = \frac{1}{4^4} = \frac{1}{256} \)
- \( e = (-5)^5 = -3125 \)
- \( f = 6^{-6} = \frac{1}{6^6} = \frac{1}{46656} \)
- \( g = 7^{-7} = \frac{1}{7^7} \)
- \( h = (-8)^8 = 8^8 \) (which is positive)
- \( i = 9^{-9} = \frac{1}{9^9} \)
- \( j = 10^{-10} = \frac{1}{10^{10}} \)

### Step 2: Calculate \( F \)

The expression for \( F \) is:
\[ F = a \cdot |b| \cdot c \cdot |d| \cdot e \cdot |f| \cdot g \cdot |h| \cdot i \cdot |j| \]

Substituting the values:
\[ F = (-1) \cdot \left|\frac{1}{4}\right| \cdot (-27) \cdot \left|\frac{1}{256}\right| \cdot (-3125) \cdot \left|\frac{1}{46656}\right| \cdot \left(\frac{1}{7^7}\right) \cdot 8^8 \cdot \left(\frac{1}{9^9}\right) \cdot \left(\frac{1}{10^{10}}\right) \]

### Step 3: Simplify \( F \)

Calculating the absolute values:
\[ F = (-1) \cdot \frac{1}{4} \cdot (-27) \cdot \frac{1}{256} \cdot (-3125) \cdot \frac{1}{46656} \cdot \frac{1}{7^7} \cdot 8^8 \cdot \frac{1}{9^9} \cdot \frac{1}{10^{10}} \]

Now, count the negative signs:
- There are 3 negative terms: \( (-1), (-27), (-3125) \), which results in a negative product.

Thus, we can rewrite \( F \) as:
\[ F = -\left(\frac{1}{4} \cdot 27 \cdot \frac{1}{256} \cdot 3125 \cdot \frac{1}{46656} \cdot \frac{1}{7^7} \cdot 8^8 \cdot \frac{1}{9^9} \cdot \frac{1}{10^{10}}\right) \]

### Step 4: Calculate \( G \)

The expression for \( G \) is:
\[ G = a \cdot b \cdot c \cdot d \cdot e \cdot f \cdot g \cdot h \cdot i \cdot j \]

Substituting the values:
\[ G = (-1) \cdot \frac{1}{4} \cdot (-27) \cdot \frac{1}{256} \cdot (-3125) \cdot \frac{1}{46656} \cdot \frac{1}{7^7} \cdot 8^8 \cdot \frac{1}{9^9} \cdot \frac{1}{10^{10}} \]

### Step 5: Simplify \( G \)

Counting the negative signs:
- There are 3 negative terms: \( (-1), (-27), (-3125) \), which results in a negative product.

Thus, we can rewrite \( G \) as:
\[ G = -\left(\frac{1}{4} \cdot 27 \cdot \frac{1}{256} \cdot 3125 \cdot \frac{1}{46656} \cdot \frac{1}{7^7} \cdot 8^8 \cdot \frac{1}{9^9} \cdot \frac{1}{10^{10}}\right) \]

### Step 6: Compare \( F \) and \( G \)

From the simplifications, we see that:
\[ F = -G \]

### Conclusion

The correct relation is:
\[ F = -G \]