To determine which expressions use mental math to find the product of [tex]\( 4 \times 2,025 \)[/tex], we'll explore each given expression to see how they break down the multiplication.
1. [tex]\( 4 \times (2,000 + 20 + 5) \)[/tex]
This expression correctly uses the distributive property. We break [tex]\( 2,025 \)[/tex] into [tex]\( 2,000 + 20 + 5 \)[/tex]:
[tex]\[
4 \times 2,025 = 4 \times (2,000 + 20 + 5)
\][/tex]
This simplifies to:
[tex]\[
= (4 \times 2,000) + (4 \times 20) + (4 \times 5)
\][/tex]
Clearly, this uses mental math to simplify the calculation. Thus, this is a valid expression.
2. [tex]\( (4 \times 2,000) + 25 \)[/tex]
This expression is incorrect. While calculating [tex]\( 4 \times 2,000\)[/tex] is part of breaking down the product, simply adding [tex]\( 25 \)[/tex] is not correct because it does not distribute [tex]\( 4 \)[/tex] across the additional part of [tex]\( 2,025 \)[/tex]. Therefore, this should not be selected.
3. [tex]\( (4 \times 2,000) + (4 \times 25) \)[/tex]
This expression also uses the distributive property correctly by breaking [tex]\( 2,025 \)[/tex] into [tex]\( 2,000 + 25 \)[/tex]:
[tex]\[
4 \times 2,025 = (4 \times 2,000) + (4 \times 25)
\][/tex]
This distribution is valid and simplifies to:
[tex]\[
= 8,000 + 100
\][/tex]
This makes it easier to perform mentally. Hence, this is a valid expression.
4. [tex]\( 4 \times (2,000 + 25) \)[/tex]
Here, we break [tex]\( 2,025 \)[/tex] into [tex]\( 2,000 + 25 \)[/tex]:
[tex]\[
4 \times 2,025 = 4 \times (2,000 + 25)
\][/tex]
By distributing, it further simplifies to:
[tex]\[
= (4 \times 2,000) + (4 \times 25) = 8,000 + 100
\][/tex]
Therefore, this expression is also valid for mental math calculations.
5. [tex]\( (4 \times 2,000 \times 25) \)[/tex]
This expression is incorrect. It combines the multiplication in a way that does not represent the original problem [tex]\( 4 \times 2,025 \)[/tex]. Thus, it is not valid for simplifying the product using mental math.
In conclusion, the valid expressions that show how to use mental math to find the product of [tex]\( 4 \times 2,025 \)[/tex] are:
[tex]\[
\begin{array}{l}
4 \times (2,000 + 20 + 5) \\
(4 \times 2,000) + (4 \times 25) \\
4 \times (2,000 + 25)
\end{array}
\][/tex]
