Answer :
Alright, let's go through each part step-by-step, paying close attention to the properties and principles of math.
(a) This exercise uses the commutative property of multiplication, which states that the order in which you multiply numbers does not change the product.
[tex]\[ 2485 \times 2346 = 2346 \times 2485 \][/tex]
Thus, this blank is already filled correctly with no additional calculation needed.
(b) Similar to the first one, this again uses the commutative property of multiplication.
[tex]\[ 2947 \times 5508 = 5508 \times 2947 \][/tex]
This blank is also correctly filled using the commutative property.
(c) This part may be trying to illustrate the identity property, but there seems to be a misunderstanding since \(3772 \times 3772\) should give a product, not just \(3772\). If correcting the right equality, it should state:
[tex]\[ 3772 \times 1 = 3772 \][/tex]
But as it stands, for \( 3772 \times 3772 \):
[tex]\[ 3772 \times 1^2 = 3772 \][/tex]
Thus, we recognize the error and correct the underlying intent:
[tex]\[ 3772 \times 1 = 3772 \][/tex]
(d) The multiplication property of zero states that any number multiplied by zero equals zero.
[tex]\[ 5358 \times 0 = 0 \][/tex]
This is correct and shows the zero property of multiplication.
(e) Distributive property allows distribution of multiplication over addition:
[tex]\[ 65 \times(100+37) = (65 \times 100) + (65 \times \square) \][/tex]
To find the missing value:
[tex]\[ 65 \times 37 = 65 \times \text{the missing number} \][/tex]
Thus, the blank should be filled with:
[tex]\[ \square = 37 \][/tex]
(f) Using the distributive property again:
[tex]\[ 256 \times (1000 + 48) = (256 \times 1000) + (256 \times \square) \][/tex]
The missing value here is:
[tex]\[ \square = 48 \][/tex]
(g) Associative property states that the way in which numbers are grouped when multiplying does not change their product:
[tex]\[ 480 \times (175 \times 903) = (480 \times 175) \times \square \][/tex]
Missing in the blank is:
[tex]\[ \square = 903 \][/tex]
(h) For the associative property with three groups:
[tex]\[ (\square \times (2030 \times 875)) = (2460 \times 2030) \times 875 \][/tex]
Thus, the missing:
[tex]\[ \square = 2460 \][/tex]
Summarizing all components:
(e) \( \square = 37 \)
(f) \( \square = 1000, \quad \square = 48 \)
(g) \( \square = 903 \)
(h) \( \square = 2460 \)
Here are the complete statements:
(e) \( 65 \times (100 + 37) = (65 \times 100) + (65 \times 37) \)
(f) \( 256 \times (1000 + 48) = (256 \times 1000) + (256 \times 48) \)
(g) \( 480 \times (175 \times 903) = (480 \times 175) \times 903 \)
(h) [tex]\( 2460 \times (2030 \times 875) = (2460 \times 2030) \times 875 \)[/tex]
(a) This exercise uses the commutative property of multiplication, which states that the order in which you multiply numbers does not change the product.
[tex]\[ 2485 \times 2346 = 2346 \times 2485 \][/tex]
Thus, this blank is already filled correctly with no additional calculation needed.
(b) Similar to the first one, this again uses the commutative property of multiplication.
[tex]\[ 2947 \times 5508 = 5508 \times 2947 \][/tex]
This blank is also correctly filled using the commutative property.
(c) This part may be trying to illustrate the identity property, but there seems to be a misunderstanding since \(3772 \times 3772\) should give a product, not just \(3772\). If correcting the right equality, it should state:
[tex]\[ 3772 \times 1 = 3772 \][/tex]
But as it stands, for \( 3772 \times 3772 \):
[tex]\[ 3772 \times 1^2 = 3772 \][/tex]
Thus, we recognize the error and correct the underlying intent:
[tex]\[ 3772 \times 1 = 3772 \][/tex]
(d) The multiplication property of zero states that any number multiplied by zero equals zero.
[tex]\[ 5358 \times 0 = 0 \][/tex]
This is correct and shows the zero property of multiplication.
(e) Distributive property allows distribution of multiplication over addition:
[tex]\[ 65 \times(100+37) = (65 \times 100) + (65 \times \square) \][/tex]
To find the missing value:
[tex]\[ 65 \times 37 = 65 \times \text{the missing number} \][/tex]
Thus, the blank should be filled with:
[tex]\[ \square = 37 \][/tex]
(f) Using the distributive property again:
[tex]\[ 256 \times (1000 + 48) = (256 \times 1000) + (256 \times \square) \][/tex]
The missing value here is:
[tex]\[ \square = 48 \][/tex]
(g) Associative property states that the way in which numbers are grouped when multiplying does not change their product:
[tex]\[ 480 \times (175 \times 903) = (480 \times 175) \times \square \][/tex]
Missing in the blank is:
[tex]\[ \square = 903 \][/tex]
(h) For the associative property with three groups:
[tex]\[ (\square \times (2030 \times 875)) = (2460 \times 2030) \times 875 \][/tex]
Thus, the missing:
[tex]\[ \square = 2460 \][/tex]
Summarizing all components:
(e) \( \square = 37 \)
(f) \( \square = 1000, \quad \square = 48 \)
(g) \( \square = 903 \)
(h) \( \square = 2460 \)
Here are the complete statements:
(e) \( 65 \times (100 + 37) = (65 \times 100) + (65 \times 37) \)
(f) \( 256 \times (1000 + 48) = (256 \times 1000) + (256 \times 48) \)
(g) \( 480 \times (175 \times 903) = (480 \times 175) \times 903 \)
(h) [tex]\( 2460 \times (2030 \times 875) = (2460 \times 2030) \times 875 \)[/tex]
