Certainly! Let's work through the given problems step-by-step to verify the identities.
### Part (a)
We need to prove that:
[tex]\[ 28 \times (5 + 14) = (28 \times 5) + (28 \times 14) \][/tex]
1. Calculate the left-hand side (LHS):
[tex]\( 28 \times (5 + 14) \)[/tex].
- Add the numbers inside the parentheses:
[tex]\[ 5 + 14 = 19 \][/tex]
- Now multiply 28 by the sum:
[tex]\[ 28 \times 19 = 532 \][/tex]
Thus, the left-hand side (LHS) is:
[tex]\[ 28 \times (5 + 14) = 532 \][/tex]
2. Calculate the right-hand side (RHS):
[tex]\((28 \times 5) + (28 \times 14)\)[/tex].
- First, multiply 28 by 5:
[tex]\[ 28 \times 5 = 140 \][/tex]
- Then, multiply 28 by 14:
[tex]\[ 28 \times 14 = 392 \][/tex]
- Finally, add the results:
[tex]\[ 140 + 392 = 532 \][/tex]
Thus, the right-hand side (RHS) is:
[tex]\[ (28 \times 5) + (28 \times 14) = 532 \][/tex]
3. Compare the results:
Since both the left-hand side and the right-hand side equal 532, we have:
[tex]\[ 28 \times (5 + 14) = (28 \times 5) + (28 \times 14) \][/tex]
### Part (b)
We need to prove that:
[tex]\[ 43 \times (25 - 10) = (43 \times 25) - (43 \times 10) \][/tex]
1. Calculate the left-hand side (LHS):
[tex]\( 43 \times (25 - 10) \)[/tex].
- Subtract the numbers inside the parentheses:
[tex]\[ 25 - 10 = 15 \][/tex]
- Now multiply 43 by the difference:
[tex]\[ 43 \times 15 = 645 \][/tex]
Thus, the left-hand side (LHS) is:
[tex]\[ 43 \times (25 - 10) = 645 \][/tex]
2. Calculate the right-hand side (RHS):
[tex]\((43 \times 25) - (43 \times 10)\)[/tex].
- First, multiply 43 by 25:
[tex]\[ 43 \times 25 = 1075 \][/tex]
- Then, multiply 43 by 10:
[tex]\[ 43 \times 10 = 430 \][/tex]
- Finally, subtract the results:
[tex]\[ 1075 - 430 = 645 \][/tex]
Thus, the right-hand side (RHS) is:
[tex]\[ (43 \times 25) - (43 \times 10) = 645 \][/tex]
3. Compare the results:
Since both the left-hand side and the right-hand side equal 645, we have:
[tex]\[ 43 \times (25 - 10) = (43 \times 25) - (43 \times 10) \][/tex]
By completing these steps, we've demonstrated the required equalities:
(a) [tex]\[ 28 \times (5 + 14) = (28 \times 5) + (28 \times 14) \][/tex]
(b) [tex]\[ 43 \times (25 - 10) = (43 \times 25) - (43 \times 10) \][/tex]
Both identities hold true!
