Certainly!
### Problem 29: [tex]\( 7 \times 6.1 \)[/tex]
To solve [tex]\( 7 \times 6.1 \)[/tex] using the distributive property and mental math, we can break down 6.1 into simpler parts:
[tex]\[ 6.1 = 6 + 0.1 \][/tex]
Now distribute the 7:
[tex]\[ 7 \times 6.1 = 7 \times (6 + 0.1) \][/tex]
[tex]\[ = (7 \times 6) + (7 \times 0.1) \][/tex]
Calculate each part:
[tex]\[ 7 \times 6 = 42 \][/tex]
[tex]\[ 7 \times 0.1 = 0.7 \][/tex]
Add these results together:
[tex]\[ 42 + 0.7 = 42.7 \][/tex]
Therefore, the product is [tex]\( 42.7 \)[/tex]. The correct choice is [tex]\( \text{d. } 42.7 \)[/tex].
### Problem 30: [tex]\( 4 \times 51^2 \)[/tex]
To solve [tex]\( 4 \times 51^2 \)[/tex], let's first compute [tex]\( 51^2 \)[/tex]:
[tex]\[ 51^2 = 51 \times 51 \][/tex]
Using the distributive property, we can simplify [tex]\( 51 \)[/tex] as:
[tex]\[ 51 = 50 + 1 \][/tex]
So,
[tex]\[ 51^2 = (50 + 1)^2 \][/tex]
[tex]\[ = 50^2 + 2 \times 50 \times 1 + 1^2 \][/tex]
[tex]\[ = 2500 + 100 + 1 \][/tex]
[tex]\[ = 2601 \][/tex]
Now multiply by 4:
[tex]\[ 4 \times 51^2 = 4 \times 2601 \][/tex]
[tex]\[ = 4 \times 2600 + 4 \times 1 \][/tex]
[tex]\[ = 10400 + 4 \][/tex]
[tex]\[ = 10404 \][/tex]
Therefore, the product is [tex]\( 10404 \)[/tex].
