What is the value of `((4.3)^(3)+(3.2)^(3)-(7.5)^(3)+10(12.9)(1.6)(1.5))/(6.5(3.6)^(3)+(1.2)^(2)-(2.8)^(2))`

Options

A. 0
B. 1
C. `0.1`
D. None of these

Answer

To solve the expression

\[ \frac{(4.3)^3 + (3.2)^3 - (7.5)^3 + 10(12.9)(1.6)(1.5)}{6.5(3.6)^3 + (1.2)^2 - (2.8)^2} \]

we will break it down step by step.

### Step 1: Calculate the numerator

The numerator is

\[ (4.3)^3 + (3.2)^3 - (7.5)^3 + 10(12.9)(1.6)(1.5) \]

We can use the identity \( a^3 + b^3 - c^3 + 3abc = (a + b - c)(a^2 + b^2 + c^2 - ab - ac - bc) \).

Let \( a = 4.3 \), \( b = 3.2 \), and \( c = 7.5 \).

### Step 2: Calculate \( a + b - c \)

\[ a + b - c = 4.3 + 3.2 - 7.5 = 7.5 - 7.5 = 0 \]

### Step 3: Calculate \( 10(12.9)(1.6)(1.5) \)

Now calculate the additional term:

\[ 10(12.9)(1.6)(1.5) = 10 \times 12.9 \times 1.6 \times 1.5 \]

Calculating this step-by-step:

1. \( 12.9 \times 1.6 = 20.64 \)
2. \( 20.64 \times 1.5 = 30.96 \)
3. \( 10 \times 30.96 = 309.6 \)

### Step 4: Substitute back into the numerator

Now, substituting back, we have:

\[ (4.3)^3 + (3.2)^3 - (7.5)^3 + 309.6 \]

Since \( a + b - c = 0 \), we can conclude that:

\[ (4.3)^3 + (3.2)^3 - (7.5)^3 = 0 \]

Thus, the numerator simplifies to:

\[ 0 + 309.6 = 309.6 \]

### Step 5: Calculate the denominator

Now, we calculate the denominator:

\[ 6.5(3.6)^3 + (1.2)^2 - (2.8)^2 \]

Calculating each term:

1. \( (3.6)^3 = 46.656 \)
2. \( (1.2)^2 = 1.44 \)
3. \( (2.8)^2 = 7.84 \)

Now substituting these values into the denominator:

\[ 6.5 \times 46.656 + 1.44 - 7.84 \]

Calculating \( 6.5 \times 46.656 \):

\[ 6.5 \times 46.656 = 303.264 \]

Now substituting back:

\[ 303.264 + 1.44 - 7.84 = 303.264 - 6.4 = 296.864 \]

### Step 6: Final Calculation

Now we have:

\[ \frac{309.6}{296.864} \]

Calculating this gives approximately:

\[ 1.041 \]

### Final Answer

Thus, the value of the expression is approximately \( 1.041 \).

---