Certainly! Let's make [tex]\( x \)[/tex] the subject of the given equation:
[tex]\[ q = \frac{1}{m-n} \left( \frac{p x}{b} + c \right)^{\frac{1}{4}} \][/tex]
Here's the step-by-step solution:
1. Isolate the fractional power term:
First, multiply both sides of the equation by [tex]\((m - n)\)[/tex]:
[tex]\[
q (m - n) = \left( \frac{p x}{b} + c \right)^{\frac{1}{4}}
\][/tex]
2. Eliminate the fourth root:
Raise both sides of the equation to the power of 4 to get rid of the fourth root:
[tex]\[
\left( q (m - n) \right)^4 = \frac{p x}{b} + c
\][/tex]
3. Simplify the left side:
Calculate the left side of the equation:
[tex]\[
\left( q (m - n) \right)^4 = q^4 (m - n)^4
\][/tex]
4. Isolate the term containing [tex]\( x \)[/tex]:
Subtract [tex]\( c \)[/tex] from both sides:
[tex]\[
q^4 (m - n)^4 - c = \frac{p x}{b}
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Multiply both sides by [tex]\( b \)[/tex] to isolate [tex]\( x \)[/tex]:
[tex]\[
b \left( q^4 (m - n)^4 - c \right) = p x
\][/tex]
6. Divide by [tex]\( p \)[/tex]:
Finally, divide both sides by [tex]\( p \)[/tex] to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{b \left( q^4 (m - n)^4 - c \right)}{p}
\][/tex]
So, the solution for [tex]\( x \)[/tex] is:
[tex]\[
x = -\frac{b \left( c - q^4 (m - n)^4 \right)}{p}
\][/tex]
Note that the negative sign appears as we rearranged the terms. Thus, the final expression for [tex]\( x \)[/tex] is:
[tex]\[
x = -\frac{b \left( c - q^4 (m - n)^4 \right)}{p}
\][/tex]
