To solve the equation [tex]\( 100 + (x + 100)^{\frac{5}{4}} = -143 \)[/tex], we will follow a systematic approach to isolate [tex]\( x \)[/tex] and solve for it. Here are the steps in detail:
1. Move the constant term to the other side of the equation:
[tex]\[
100 + (x + 100)^{\frac{5}{4}} = -143
\][/tex]
Subtract 100 from both sides:
[tex]\[
(x + 100)^{\frac{5}{4}} = -143 - 100
\][/tex]
Simplify the right-hand side:
[tex]\[
(x + 100)^{\frac{5}{4}} = -243
\][/tex]
2. Solve for [tex]\( x \)[/tex] by isolating the term with [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], recognize that taking the 4/5th power (the inverse of the 5/4th power) on both sides will help us remove the exponent on the left side. However, note that raising a negative number to a fractional exponent generally results in complex numbers due to the roots involved.
Using complex numbers, take both the principal 4/5th power of [tex]\(-243\)[/tex]:
[tex]\[
x + 100 = \left( -243 \right)^{\frac{4}{5}}
\][/tex]
This value involves complex numbers because of the negative base and the fractional exponent.
3. Express the complex number solution:
Utilize the properties of complex exponentiation and roots to solve:
When performing these steps, the result is:
[tex]\[
x + 100 = -165.530376544371 \pm 47.6106054356903i
\][/tex]
4. Isolate [tex]\( x \)[/tex]:
Subtract 100 from both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = -165.530376544371 \pm 47.6106054356903i - 100
\][/tex]
Simplify the real part:
[tex]\[
x = -165.530376544371 - 100 \pm 47.6106054356903i
\][/tex]
Therefore, we have:
[tex]\[
x = -265.530376544371 \pm 47.6106054356903i
\][/tex]
Thus, the two solutions are:
[tex]\[
\boxed{-265.530376544371 - 47.6106054356903i}
\][/tex]
and
[tex]\[
\boxed{-265.530376544371 + 47.6106054356903i}
\][/tex]
