To find the product of [tex]\( 8 \times 490 \)[/tex] using mental math, we can use various properties of multiplication, such as the distributive property. Let's analyze each given expression to see which ones correctly use these properties to simplify the calculation.
1. Expression: [tex]\( 8 + (400 \times 90) \)[/tex]
This expression is incorrect. It does not correctly split 490 and adds 8 to the product of 400 and 90, which is not related to [tex]\( 8 \times 490 \)[/tex].
2. Expression: [tex]\( (8 \times 400) + (8 \times 90) \)[/tex]
This expression is correct. By using the distributive property, we can break down 490 into 400 and 90:
[tex]\[
490 = 400 + 90.
\][/tex]
Thus,
[tex]\[
8 \times 490 = 8 \times (400 + 90) = (8 \times 400) + (8 \times 90).
\][/tex]
3. Expression: [tex]\( (8 \times 400) + (8 \times 9) \)[/tex]
This expression is incorrect. It splits 90 into 9, instead of correctly breaking down 490.
4. Expression: [tex]\( (8 \times 500) - (8 \times 10) \)[/tex]
This expression is correct. By using the distributive property again, we can break down 490 into 500 and subtract 10:
[tex]\[
490 = 500 - 10.
\][/tex]
Thus,
[tex]\[
8 \times 490 = 8 \times (500 - 10) = (8 \times 500) - (8 \times 10).
\][/tex]
5. Expression: [tex]\( 8 \times (500 \times 10) \)[/tex]
This expression is incorrect. It implies multiplying 8 by 5000, which does not relate to [tex]\( 8 \times 490 \)[/tex].
Therefore, the expressions that correctly show how to use mental math to find the product of [tex]\( 8 \times 490 \)[/tex] are:
[tex]\[ (8 \times 400) + (8 \times 90) \][/tex]
[tex]\[ (8 \times 500) - (8 \times 10) \][/tex]
So, the correct selections are:
[tex]\[ \boxed{2, 4} \][/tex]
