What is the value of `((1.2)^(3)+(0.8)^(3)+(0.7)^(3)-2.016)/((1.35)[(1.2)^(2)+(0.8)^(2)+(0.7)^(2)-0.96-0.84-0.56])`?

Options

A. `(1)/(4)`
B. `(1)/(2)`
C. `1`
D. `2`

Answer

To solve the given problem step by step, we will first rewrite the expression and then simplify it.

### Step 1: Rewrite the expression
The expression we need to evaluate is:

\[ \frac{(1.2)^3 + (0.8)^3 + (0.7)^3 - 2.016}{1.35 \cdot ((1.2)^2 + (0.8)^2 + (0.7)^2 - 0.96 - 0.84 - 0.56)} \]

### Step 2: Calculate the numerator
First, we will calculate the cubes of the numbers in the numerator:

\[ (1.2)^3 = 1.728, \quad (0.8)^3 = 0.512, \quad (0.7)^3 = 0.343 \]

Now, sum these values:

\[ 1.728 + 0.512 + 0.343 = 2.583 \]

Now, subtract 2.016 from this sum:

\[ 2.583 - 2.016 = 0.567 \]

So, the numerator simplifies to \(0.567\).

### Step 3: Calculate the denominator
Next, we will calculate the squares of the numbers in the denominator:

\[ (1.2)^2 = 1.44, \quad (0.8)^2 = 0.64, \quad (0.7)^2 = 0.49 \]

Now, sum these values:

\[ 1.44 + 0.64 + 0.49 = 2.57 \]

Now, subtract \(0.96\), \(0.84\), and \(0.56\) from this sum:

\[ 2.57 - 0.96 - 0.84 - 0.56 = 2.57 - 2.36 = 0.21 \]

Now, multiply this result by \(1.35\):

\[ 1.35 \cdot 0.21 = 0.2835 \]

### Step 4: Combine the results
Now we can combine the results from the numerator and denominator:

\[ \frac{0.567}{0.2835} \]

### Step 5: Calculate the final value
Now, we will perform the division:

\[ \frac{0.567}{0.2835} \approx 2 \]

Thus, the value of the expression is approximately \(2\).

### Final Answer
The value of the expression is \(2\).

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