Value of $\frac{986 imes 986 imes 986 - 742 imes 742 imes 742}{986 imes 986 + 986 imes 742 + 742 imes 742}$ is:

Q: Value of $\frac{986 \times 986 \times 986 - 742 \times 742 \times 742}{986 \times 986 + 986 \times 742 + 742 \times 742}$ is:

To solve the given expression $\frac{986 \times 986 \times 986 - 742 \times 742 \times 742}{986 \times 986 + 986 \times 742 + 742 \times 742}$, we can use the identity for the difference of cubes: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \] In this case, let $ a = 986 $ and $ b = 742 $. Applying the identity: \[ a^3 - b^3 = (986 - 742)(986^2 + 986 \times 742 + 742^2) \] The denominator of the given expression is exactly $ 986^2 + 986 \times 742 + 742^2 $, which matches the second part of the identity. Thus, the expression simplifies to: \[ \frac{(986 - 742)(986^2 + 986 \times 742 + 742^2)}{986^2 + 986 \times 742 + 742^2} \] The terms $ 986^2 + 986 \times 742 + 742^2 $ in the numerator and denominator cancel each other out, leaving: \[ 986 - 742 = 244 \] Therefore, the value of the expression is 244, which corresponds to Option 4. **Explanation for why other options are incorrect:** 1. **Option 1 (1044):** This value is incorrect because the calculation of $ 986 - 742 $ results in 244, not 1044. The difference of cubes identity simplifies the expression to a simple subtraction, not a larger number like 1044. 2. **Option 2 (344):** This value is incorrect for the same reason. The subtraction $ 986 - 742 $ results in 244, not 344. The identity used ensures that the expression simplifies directly to the difference between the two numbers. 3. **Option 3 (144):** This value is incorrect because the subtraction $ 986 - 742 $ results in 244, not 144. The identity clearly shows that the expression simplifies to the difference of the two numbers, which is 244. In conclusion, Option 4 (244) is the correct answer as it accurately reflects the result of the simplified expression using the difference of cubes identity.

Question

Value of $\frac{986 	imes 986 	imes 986 - 742 	imes 742 	imes 742}{986 	imes 986 + 986 	imes 742 + 742 	imes 742}$ is:

Accepted Answer

To solve the given expression $\frac{986 \times 986 \times 986 - 742 \times 742 \times 742}{986 \times 986 + 986 \times 742 + 742 \times 742}$, we can use the identity for the difference of cubes:

$$ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $$

In this case, let $ a = 986 $ and $ b = 742 $. Applying the identity:

$$ a^3 - b^3 = (986 - 742)(986^2 + 986 \times 742 + 742^2) $$

The denominator of the given expression is exactly $ 986^2 + 986 \times 742 + 742^2 $, which matches the second part of the identity.

Thus, the expression simplifies to:

$$ \frac{(986 - 742)(986^2 + 986 \times 742 + 742^2)}{986^2 + 986 \times 742 + 742^2} $$

The terms $ 986^2 + 986 \times 742 + 742^2 $ in the numerator and denominator cancel each other out, leaving:

$$ 986 - 742 = 244 $$

Therefore, the value of the expression is 244, which corresponds to Option 4.

**Explanation for why other options are incorrect:**

1. **Option 1 (1044):** This value is incorrect because the calculation of $ 986 - 742 $ results in 244, not 1044. The difference of cubes identity simplifies the expression to a simple subtraction, not a larger number like 1044.

2. **Option 2 (344):** This value is incorrect for the same reason. The subtraction $ 986 - 742 $ results in 244, not 344. The identity used ensures that the expression simplifies directly to the difference between the two numbers.

3. **Option 3 (144):** This value is incorrect because the subtraction $ 986 - 742 $ results in 244, not 144. The identity clearly shows that the expression simplifies to the difference of the two numbers, which is 244.

In conclusion, Option 4 (244) is the correct answer as it accurately reflects the result of the simplified expression using the difference of cubes identity.