To determine which expressions are equivalent to multiplying 35 by 21, let’s analyze each option step-by-step.
### Option A: [tex]\((7 \times 5) \times (7 \times 3)\)[/tex]
First, evaluate the individual multiplications within the parentheses:
[tex]\[ 7 \times 5 = 35 \][/tex]
[tex]\[ 7 \times 3 = 21 \][/tex]
Now, multiply the results:
[tex]\[ 35 \times 21 \][/tex]
Since the original expression is [tex]\(35 \times 21\)[/tex], this option is equivalent.
### Option B: [tex]\((30 \times 21) + (5 \times 21)\)[/tex]
Here, let's use the distributive property to verify:
[tex]\[ (30 + 5) \times 21 \][/tex]
First, evaluate the sum:
[tex]\[ 30 + 5 = 35 \][/tex]
Then multiply:
[tex]\[ 35 \times 21 \][/tex]
This matches the original expression, so this option is equivalent.
### Option C: [tex]\(7(5 \times 3) 7(5 + 3)\)[/tex]
Let's break this down step-by-step:
1. Evaluate the expression inside the first set of parentheses:
[tex]\[ 5 \times 3 = 15 \][/tex]
2. Multiply by 7:
[tex]\[ 7 \times 15 \][/tex]
Moving to the next part of the expression:
1. Evaluate the expression inside the second set of parentheses:
[tex]\[ 5 + 3 = 8 \][/tex]
2. Multiply by 7:
[tex]\[ 7 \times 8 = 56 \][/tex]
The rewritten expression suggests comparing [tex]\(7 \times 15\)[/tex] with [tex]\(56\)[/tex]. Since neither of these products equal [tex]\(35 \times 21\)[/tex], option C is not equivalent.
### Conclusion:
The expressions that are equivalent to multiplying 35 by 21 are:
- Option A [tex]\((7 \times 5) \times (7 \times 3)\)[/tex]
- Option B [tex]\((30 \times 21) + (5 \times 21)\)[/tex]
So, the correct answers are:
- A
- B
