32 × (27 × 19) = (32 × ____) × 19
27
Idea: In multiplication, changing the grouping does not change the answer. This is called the associative property.
Property (general form):
\[ (a \times (b \times c)) = ((a \times b) \times c) \]
Match to our numbers:
\[ a = 32,\quad b = 27,\quad c = 19 \]
Apply the property:
\[ 32 \times (27 \times 19) = (32 \times 27) \times 19 \]
Now compare with the pattern on the right side:
\[ (32 \times \_\_\_\_) \times 19 \]
The blank must be the number paired with 32 inside the parentheses, which is:
27.
Quick check (optional):
\[ 27 \times 19 = 513 \]
\[ 32 \times 513 = 16416 \]
\[ 32 \times 27 = 864 \]
\[ 864 \times 19 = 16416 \]
Both sides give the same result, so the blank is correct.