Solving the Integer Expression

The problem asks us to find the value of the mathematical expression:
$(-27) \times (-16) + (-27) \times (-14)$

Method 1: Direct Calculation

First, we calculate each multiplication part separately:

• The product of two negative numbers is positive. So, $(-27) \times (-16)$.
  We can multiply 27 by 16:
  $27 \times 16 = 27 \times (10 + 6) = (27 \times 10) + (27 \times 6) = 270 + 162 = 432$.
  Therefore, $(-27) \times (-16) = 432$.
• Next, we calculate $(-27) \times (-14)$.
  Similarly, $27 \times 14 = 27 \times (10 + 4) = (27 \times 10) + (27 \times 4) = 270 + 108 = 378$.
  Therefore, $(-27) \times (-14) = 378$.

Now, we add the results of the two multiplications:

$432 + 378 = 810$

Method 2: Using the Distributive Property

We can simplify the calculation by using the distributive property of multiplication over addition, which states that a × b + a × c = a × (b + c).

In our expression, $(-27) \times (-16) + (-27) \times (-14)$, the common factor is (-27).

Applying the distributive property:

• Factor out (-27):
  $(-27) \times ((-16) + (-14))$
• Add the numbers inside the parenthesis:
  $(-16) + (-14) = -16 - 14 = -30$.
• Now, multiply (-27) by (-30):
  $(-27) \times (-30)$
• The product of two negative numbers is positive.
  $27 \times 30 = 810$.
• So, $(-27) \times (-30) = 810$.

Final Answer Calculation

Both methods lead to the same result. The value of the expression $(-27) \times (-16) + (-27) \times (-14)$ is 810.