The value of `(5.35 xx 5 .35 xx 5 .35 + 3.65 xx 3 .65 xx 3 .65)/(53 . 5 xx 53.5 + 36.5 xx 36.5 - 53.5 xx 36.5)` is :

Options

A. 9
B. 0.9
C. 90
D. 0.09

Answer

To solve the expression \((5.35 \times 5.35 \times 5.35 + 3.65 \times 3.65 \times 3.65)/(53.5 \times 53.5 + 36.5 \times 36.5 - 53.5 \times 36.5)\), we can follow these steps:

### Step 1: Recognize the Patterns
The numerator can be recognized as the sum of cubes:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
where \(a = 5.35\) and \(b = 3.65\).

### Step 2: Calculate the Numerator
Using the formula for the sum of cubes:
\[ 5.35^3 + 3.65^3 = (5.35 + 3.65)(5.35^2 - 5.35 \times 3.65 + 3.65^2) \]
Calculating \(5.35 + 3.65\):
\[ 5.35 + 3.65 = 9 \]
Now, we need to calculate \(5.35^2\), \(3.65^2\), and \(5.35 \times 3.65\).

### Step 3: Calculate \(5.35^2\), \(3.65^2\), and \(5.35 \times 3.65\)
\[ 5.35^2 = 28.6225 \]
\[ 3.65^2 = 13.3225 \]
\[ 5.35 \times 3.65 = 19.5275 \]

### Step 4: Substitute Back into the Numerator
Now substituting back:
\[ 5.35^2 - 5.35 \times 3.65 + 3.65^2 = 28.6225 - 19.5275 + 13.3225 = 22.4175 \]
Thus, the numerator becomes:
\[ 9 \times 22.4175 = 201.7575 \]

### Step 5: Calculate the Denominator
The denominator can be simplified using the identity:
\[ a^2 + b^2 - ab = (a + b)^2 - ab - ab = (a + b)(a + b - ab) \]
where \(a = 53.5\) and \(b = 36.5\).

Calculating \(53.5^2\), \(36.5^2\), and \(53.5 \times 36.5\):
\[ 53.5^2 = 2862.25 \]
\[ 36.5^2 = 1332.25 \]
\[ 53.5 \times 36.5 = 1957.75 \]

### Step 6: Substitute Back into the Denominator
Now substituting back:
\[ 53.5^2 + 36.5^2 - 53.5 \times 36.5 = 2862.25 + 1332.25 - 1957.75 = 3236.75 \]

### Step 7: Final Calculation
Now substituting the numerator and denominator into the expression:
\[ \frac{201.7575}{3236.75} \]

### Step 8: Simplify the Expression
To simplify:
\[ \frac{201.7575}{3236.75} = 0.0622 \approx 0.9 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{0.9} \]