To find the value of [tex]\( k \)[/tex] in the given equation [tex]\( (-2)^{k+1} \times \left(\frac{-1}{2}\right)^4 = (-2)^7 \)[/tex], let's proceed step-by-step.
1. Simplify the right-hand side of the equation:
[tex]\[
(-2)^7 = -128
\][/tex]
2. Simplify the left-hand side of the equation:
- First, we simplify [tex]\( \left(\frac{-1}{2}\right)^4 \)[/tex].
[tex]\[
\left(\frac{-1}{2}\right)^4 = \left(\frac{1}{2}\right)^4 = \frac{1}{16}
\][/tex]
- Thus, the left-hand side can be rewritten as:
[tex]\[
(-2)^{k+1} \times \frac{1}{16}
\][/tex]
3. Clear the fraction by multiplying both sides of the equation by 16:
[tex]\[
(-2)^{k+1} = (-2)^7 \times 16
\][/tex]
4. Calculate [tex]\( (-2)^7 \times 16 \)[/tex]:
[tex]\[
(-2)^7 \times 16 = -128 \times 16 = -2048
\][/tex]
So, we have:
[tex]\[
(-2)^{k+1} = -2048
\][/tex]
5. Rewrite [tex]\(-2048\)[/tex] as a power of [tex]\(-2\)[/tex]:
[tex]\[
-2048 = (-2)^{11}
\][/tex]
Therefore:
[tex]\[
(-2)^{k+1} = (-2)^{11}
\][/tex]
6. Since the bases are equal, set the exponents equal to each other:
[tex]\[
k+1 = 11
\][/tex]
7. Solve for [tex]\( k \)[/tex]:
[tex]\[
k = 11 - 1 = 10
\][/tex]
Hence, the value of [tex]\( k \)[/tex] is [tex]\( \boxed{10} \)[/tex].
