Question
Solve the following equations: $\sqrt{\frac{\text{a}}{\text{b}}}=\Big(\frac{\text{b}}{\text{a}}\Big)^{1-2\text{x}},$ where a, b are distinct positive primes.
✓
Answer
$\sqrt{\frac{\text{a}}{\text{b}}}=\Big(\frac{\text{b}}{\text{a}}\Big)^{1-2\text{x}}$$\Rightarrow\Big(\frac{\text{a}}{\text{b}}\Big)^{\frac{1}{2}}=\Big(\frac{\text{b}}{\text{a}}\Big)^{1-2\text{x}}$
$\Rightarrow\Big(\frac{\text{a}}{\text{b}}\Big)^{\frac{1}{2}}=\Big(\frac{\text{a}}{\text{b}}\Big)^{-1+2\text{x}}$
$\Rightarrow\frac{1}{2}=-1+2\text{x}$
$\Rightarrow2\text{x}=\frac{1}{2}+1$
$\Rightarrow2\text{x}=\frac{3}{2}$
$\Rightarrow2\text{x}=\frac{3}{4}$
$\Rightarrow\Big(\frac{\text{a}}{\text{b}}\Big)^{\frac{1}{2}}=\Big(\frac{\text{a}}{\text{b}}\Big)^{-1+2\text{x}}$
$\Rightarrow\frac{1}{2}=-1+2\text{x}$
$\Rightarrow2\text{x}=\frac{1}{2}+1$
$\Rightarrow2\text{x}=\frac{3}{2}$
$\Rightarrow2\text{x}=\frac{3}{4}$
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