Answer :
To find the value that satisfies the equation [tex]\(\sqrt{-x} = 4x^9\)[/tex], we need to follow these steps:
1. Understand the equation: The equation given is [tex]\(\sqrt{-x} = 4x^9\)[/tex]. This means the square root of [tex]\(-x\)[/tex] is equal to [tex]\(4x^9\)[/tex].
2. Square both sides: To eliminate the square root, square both sides of the equation:
[tex]\[
(-x) = (4x^9)^2
\][/tex]
3. Simplify the equation: Simplify the right side:
[tex]\[
(-x) = 16x^{18}
\][/tex]
4. Rearrange the equation: Move all terms to one side:
[tex]\[
16x^{18} + x = 0
\][/tex]
5. Factor the equation: Factor out [tex]\(x\)[/tex]:
[tex]\[
x(16x^{17} + 1) = 0
\][/tex]
6. Solve the equation: The solutions to the equation are when either factor equals zero:
- Solution 1: [tex]\(x = 0\)[/tex]
- Solution 2: Solve [tex]\(16x^{17} + 1 = 0\)[/tex]
[tex]\[
16x^{17} = -1
\][/tex]
[tex]\[
x^{17} = -\frac{1}{16}
\][/tex]
[tex]\[
x = \left(-\frac{1}{16}\right)^{\frac{1}{17}}
\][/tex]
7. Complex result: Since raising a negative number to a fractional power can result in a complex number, the solution for the equation will be complex. The result is approximately:
[tex]\[
x \approx 0.835 + 0.156i
\][/tex]
This complex number [tex]\((0.835 + 0.156i)\)[/tex] is the solution that satisfies the original equation [tex]\(\sqrt{-x} = 4x^9\)[/tex].
1. Understand the equation: The equation given is [tex]\(\sqrt{-x} = 4x^9\)[/tex]. This means the square root of [tex]\(-x\)[/tex] is equal to [tex]\(4x^9\)[/tex].
2. Square both sides: To eliminate the square root, square both sides of the equation:
[tex]\[
(-x) = (4x^9)^2
\][/tex]
3. Simplify the equation: Simplify the right side:
[tex]\[
(-x) = 16x^{18}
\][/tex]
4. Rearrange the equation: Move all terms to one side:
[tex]\[
16x^{18} + x = 0
\][/tex]
5. Factor the equation: Factor out [tex]\(x\)[/tex]:
[tex]\[
x(16x^{17} + 1) = 0
\][/tex]
6. Solve the equation: The solutions to the equation are when either factor equals zero:
- Solution 1: [tex]\(x = 0\)[/tex]
- Solution 2: Solve [tex]\(16x^{17} + 1 = 0\)[/tex]
[tex]\[
16x^{17} = -1
\][/tex]
[tex]\[
x^{17} = -\frac{1}{16}
\][/tex]
[tex]\[
x = \left(-\frac{1}{16}\right)^{\frac{1}{17}}
\][/tex]
7. Complex result: Since raising a negative number to a fractional power can result in a complex number, the solution for the equation will be complex. The result is approximately:
[tex]\[
x \approx 0.835 + 0.156i
\][/tex]
This complex number [tex]\((0.835 + 0.156i)\)[/tex] is the solution that satisfies the original equation [tex]\(\sqrt{-x} = 4x^9\)[/tex].
