Let's analyze each of the provided expressions for negative integer values of [tex]\( k \)[/tex] to determine which yields the smallest value.
Consider the possible values for [tex]\( k \)[/tex] to be [tex]\( -1, -2, -3, \)[/tex] and [tex]\( -4 \)[/tex].
### Expression A: [tex]\( k^9 - 1 \)[/tex]
- For [tex]\( k = -1 \)[/tex]: [tex]\( (-1)^9 - 1 = -1 - 1 = -2 \)[/tex]
- For [tex]\( k = -2 \)[/tex]: [tex]\( (-2)^9 - 1 = -512 - 1 = -513 \)[/tex]
- For [tex]\( k = -3 \)[/tex]: [tex]\( (-3)^9 - 1 = -19683 - 1 = -19684 \)[/tex]
- For [tex]\( k = -4 \)[/tex]: [tex]\( (-4)^9 - 1 = -262144 - 1 = -262145 \)[/tex]
So, the results for [tex]\( k^9 - 1 \)[/tex] are: [tex]\(-2, -513, -19684, -262145\)[/tex].
### Expression B: [tex]\( k^{10} \)[/tex]
- For [tex]\( k = -1 \)[/tex]: [tex]\( (-1)^{10} = 1 \)[/tex]
- For [tex]\( k = -2 \)[/tex]: [tex]\( (-2)^{10} = 1024 \)[/tex]
- For [tex]\( k = -3 \)[/tex]: [tex]\( (-3)^{10} = 59049 \)[/tex]
- For [tex]\( k = -4 \)[/tex]: [tex]\( (-4)^{10} = 1048576 \)[/tex]
So, the results for [tex]\( k^{10} \)[/tex] are: [tex]\(1, 1024, 59049, 1048576\)[/tex].
### Expression C: [tex]\( -1000^k \)[/tex]
- For [tex]\( k = -1 \)[/tex]: [tex]\( -1000^{-1} = -\frac{1}{1000} = -0.001 \)[/tex]
- For [tex]\( k = -2 \)[/tex]: [tex]\( -1000^{-2} = -\frac{1}{1000000} = -0.000001 \)[/tex]
- For [tex]\( k = -3 \)[/tex]: [tex]\( -1000^{-3} = -\frac{1}{1000000000} = -0.000000001 \)[/tex]
- For [tex]\( k = -4 \)[/tex]: [tex]\( -1000^{-4} = -\frac{1}{1000000000000} = -0.000000000001 \)[/tex]
So, the results for [tex]\( -1000^k \)[/tex] are: [tex]\(-0.001, -0.000001, -0.000000001, -0.000000000001\)[/tex].
### Expression D: [tex]\( -k^7 \)[/tex]
- For [tex]\( k = -1 \)[/tex]: [tex]\( -(-1)^7 = -( -1 ) = 1 \)[/tex]
- For [tex]\( k = -2 \)[/tex]: [tex]\( -(-2)^7 = -(-128) = 128 \)[/tex]
- For [tex]\( k = -3 \)[/tex]: [tex]\( -(-3)^7 = -(-2187) = 2187 \)[/tex]
- For [tex]\( k = -4 \)[/tex]: [tex]\( -(-4)^7 = -(-16384) = 16384 \)[/tex]
So, the results for [tex]\( -k^7 \)[/tex] are: [tex]\(1, 128, 2187, 16384\)[/tex].
### Determining the Smallest Value
Now, we need to find the smallest value among all the results for each expression:
- The smallest value for [tex]\( k^9 - 1 \)[/tex] is [tex]\(-262145\)[/tex].
- The smallest value for [tex]\( k^{10} \)[/tex] is [tex]\(1\)[/tex].
- The smallest value for [tex]\( -1000^k \)[/tex] is [tex]\(-0.001\)[/tex].
- The smallest value for [tex]\( -k^7 \)[/tex] is [tex]\(1\)[/tex].
Comparing [tex]\(-262145, 1, -0.001, \)[/tex] and [tex]\(1\)[/tex], the smallest value is [tex]\(-262145\)[/tex].
Thus, the expression [tex]\( k^9 - 1 \)[/tex] yields the smallest value when [tex]\( k \)[/tex] is a negative integer. The correct answer is [tex]\( \boxed{A} \)[/tex].
