Let's evaluate each of the given mathematical expressions step-by-step.
### Expression (a): [tex]\( a - (b++)^{-c} \)[/tex]
Given: [tex]\( a = 2 \)[/tex], [tex]\( b = 3 \)[/tex], [tex]\( c = 9 \)[/tex].
1. Evaluate [tex]\(b++\)[/tex]:
- The expression [tex]\(b++\)[/tex] typically means using [tex]\(b\)[/tex] before incrementing it. So, initially, [tex]\(b = 3\)[/tex].
- After using [tex]\(b\)[/tex] in this expression, [tex]\(b\)[/tex] is incremented by 1, so [tex]\(b\)[/tex] becomes [tex]\(4\)[/tex].
2. Evaluate [tex]\((b++)^{-c}\)[/tex]:
- Since [tex]\(b++\)[/tex] used the value [tex]\(b = 3\)[/tex], we take [tex]\(3\)[/tex] and raise it to the power of [tex]\(-9\)[/tex]:
- [tex]\((3)^{-9}\)[/tex] which is the same as [tex]\(\frac{1}{3^9}\)[/tex].
- Calculating [tex]\(3^9\)[/tex]:
- [tex]\(3^9 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 19683\)[/tex].
- Therefore, [tex]\((3)^{-9} = \frac{1}{19683}\)[/tex].
3. Subtract this from [tex]\(a\)[/tex]:
- So, [tex]\(a - \frac{1}{19683}\)[/tex]:
- [tex]\(2 - \frac{1}{19683}\)[/tex].
- This evaluates to [tex]\(1.9999491947365746\)[/tex].
Hence, the value of expression (a) is approximately [tex]\(1.9999491947365746\)[/tex].
### Expression (b): [tex]\( a \cdot (++b) \% c \)[/tex]
Given: [tex]\( a = 2 \)[/tex], [tex]\( b = 4 \)[/tex] (from the previous increment), [tex]\( c = 9 \)[/tex].
1. Evaluate [tex]\(++b\)[/tex]:
- The expression [tex]\(++b\)[/tex] means increment [tex]\(b\)[/tex] by 1 before using it.
- So, [tex]\(b\)[/tex] which was [tex]\(4\)[/tex] is incremented to [tex]\(5\)[/tex].
2. Multiply [tex]\(a\)[/tex] with the new [tex]\(b\)[/tex]:
- Multiply [tex]\(a = 2\)[/tex] by [tex]\(b = 5\)[/tex]:
- [tex]\(2 \cdot 5 = 10\)[/tex].
3. Find the result modulo [tex]\(c\)[/tex]:
- Find the remainder when [tex]\(10\)[/tex] is divided by [tex]\(9\)[/tex]:
- [tex]\(10 \% 9 = 1\)[/tex].
Hence, the value of expression (b) is [tex]\(1\)[/tex].
### Summary
- Expression (a): [tex]\(1.9999491947365746\)[/tex]
- Expression (b): [tex]\(1\)[/tex]
