The value of : `(427 xx 427 xx 427 + 325 xx 325 xx 325)/(42.7 xx 42.7 + 32.5 xx 32.5 - 42.7 xx 32.5)`

Options

A. 75.2
B. 752
C. 75200
D. 7520

Answer

To solve the expression \((427 \times 427 \times 427 + 325 \times 325 \times 325) / (42.7 \times 42.7 + 32.5 \times 32.5 - 42.7 \times 32.5)\), we can follow these steps:

### Step 1: Simplify the Numerator
The numerator is \(427^3 + 325^3\). We can use the identity for the sum of cubes:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Here, let \(a = 427\) and \(b = 325\).

Calculating \(a + b\):
\[ 427 + 325 = 752 \]

Calculating \(a^2\):
\[ 427^2 = 182929 \]

Calculating \(b^2\):
\[ 325^2 = 105625 \]

Calculating \(ab\):
\[ 427 \times 325 = 138775 \]

Now substituting into the identity:
\[ 427^3 + 325^3 = (752)(182929 - 138775 + 105625) \]

Calculating \(182929 - 138775 + 105625\):
\[ 182929 - 138775 = 44154 \]
\[ 44154 + 105625 = 149779 \]

Thus, the numerator becomes:
\[ 427^3 + 325^3 = 752 \times 149779 \]

### Step 2: Simplify the Denominator
The denominator is \(42.7^2 + 32.5^2 - 42.7 \times 32.5\). We can use the identity:
\[ a^2 + b^2 - ab = \frac{1}{2}((a - b)^2 + a^2 + b^2) \]
Here, let \(a = 42.7\) and \(b = 32.5\).

Calculating \(a^2\):
\[ 42.7^2 = 1825.29 \]

Calculating \(b^2\):
\[ 32.5^2 = 1056.25 \]

Calculating \(ab\):
\[ 42.7 \times 32.5 = 1383.75 \]

Now substituting into the expression:
\[ 42.7^2 + 32.5^2 - 42.7 \times 32.5 = 1825.29 + 1056.25 - 1383.75 \]

Calculating:
\[ 1825.29 + 1056.25 = 2881.54 \]
\[ 2881.54 - 1383.75 = 1497.79 \]

### Step 3: Combine the Results
Now we can combine the results from the numerator and denominator:
\[ \frac{752 \times 149779}{1497.79} \]

### Step 4: Calculate the Final Value
To find the value, we can perform the division:
\[ \frac{752 \times 149779}{1497.79} \approx 752 \times 100.02 \approx 75200 \]

### Final Answer
Thus, the value of the expression is approximately \(75200\).

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