To solve the equation [tex]\( v^3 = -17 \)[/tex], we need to find the value of [tex]\( v \)[/tex] such that when [tex]\( v \)[/tex] is cubed, the result is [tex]\(-17\)[/tex].
Let's start by isolating [tex]\( v \)[/tex].
First, we want to express [tex]\( v \)[/tex] in terms of [tex]\(-17\)[/tex]:
[tex]\[
v = (-17)^{1/3}
\][/tex]
This expression [tex]\( (-17)^{1/3} \)[/tex] represents the cube root of [tex]\(-17\)[/tex]. We want to determine what [tex]\( v \)[/tex] is when taking the cube root of [tex]\(-17\)[/tex].
It's important to note that the cube root of a negative number can involve complex numbers, since real numbers only. In the context of complex numbers, we find:
[tex]\[
v = (1.2856407953291178 + 2.2267951777932917i)
\][/tex]
This solution is complex, meaning there is no real solution to the equation [tex]\( v^3 = -17 \)[/tex]. Therefore, we conclude:
[tex]\[
v = \text{No solution}
\][/tex]
