Answer :
Sure! Let's solve the equation step by step:
The given equation is:
[tex]\[
-2x^6 = 128
\][/tex]
1. Isolate [tex]\( x^6 \)[/tex]:
To simplify the equation, first divide both sides by [tex]\(-2\)[/tex] to solve for [tex]\( x^6 \)[/tex]:
[tex]\[
x^6 = \frac{128}{-2}
\][/tex]
This gives:
[tex]\[
x^6 = -64
\][/tex]
2. Determine the Nature of the Solution:
We need to find the values of [tex]\( x \)[/tex] that satisfy [tex]\( x^6 = -64 \)[/tex]. Notice that raising a real number to an even power results in a non-negative number. This implies that [tex]\( -64 \)[/tex] cannot be attained with any real number raised to the sixth power.
3. Complex Solutions:
Since real numbers cannot satisfy this equation, we will consider complex numbers. Evaluating complex roots typically involves using methods that account for both real and imaginary parts.
In conclusion, the equation [tex]\( x^6 = -64 \)[/tex] does not have real solutions. It has complex solutions, which are derived mathematically in specialized methods. Since real numbers raised to the sixth power cannot be negative, any solution for this equation must involve complex numbers.
The given equation is:
[tex]\[
-2x^6 = 128
\][/tex]
1. Isolate [tex]\( x^6 \)[/tex]:
To simplify the equation, first divide both sides by [tex]\(-2\)[/tex] to solve for [tex]\( x^6 \)[/tex]:
[tex]\[
x^6 = \frac{128}{-2}
\][/tex]
This gives:
[tex]\[
x^6 = -64
\][/tex]
2. Determine the Nature of the Solution:
We need to find the values of [tex]\( x \)[/tex] that satisfy [tex]\( x^6 = -64 \)[/tex]. Notice that raising a real number to an even power results in a non-negative number. This implies that [tex]\( -64 \)[/tex] cannot be attained with any real number raised to the sixth power.
3. Complex Solutions:
Since real numbers cannot satisfy this equation, we will consider complex numbers. Evaluating complex roots typically involves using methods that account for both real and imaginary parts.
In conclusion, the equation [tex]\( x^6 = -64 \)[/tex] does not have real solutions. It has complex solutions, which are derived mathematically in specialized methods. Since real numbers raised to the sixth power cannot be negative, any solution for this equation must involve complex numbers.
