Solve the equation y = sqrt(log(x) / e^(x^2))
Understand the Problem
The question provides an equation and likely asks to solve or simplify it. The equation involves logarithmic and exponential functions along with a square root.
Answer
$ y = \frac{\sqrt{\log x}}{e^{\frac{x^2}{2}}} $
Answer for screen readers
$$ y = \frac{\sqrt{\log x}}{e^{\frac{x^2}{2}}} $$
Steps to Solve
- Rewrite the square root as a power of 1/2
We can rewrite the square root as a power of $1/2$ as follows: $$ y = \left(\frac{\log x}{e^{x^2}}\right)^{\frac{1}{2}} $$
- Separate the power of 1/2
Apply the power to both the numerator and the denominator:
$$ y = \frac{(\log x)^{\frac{1}{2}}}{(e^{x^2})^{\frac{1}{2}}} $$
- Simplify the denominator
We simplify the denominator by multiplying the exponents:
$$ y = \frac{(\log x)^{\frac{1}{2}}}{e^{\frac{x^2}{2}}} $$
- Rewrite the numerator with a square root
Rewrite $(\log x)^{\frac{1}{2}}$ as $\sqrt{\log x}$:
$$ y = \frac{\sqrt{\log x}}{e^{\frac{x^2}{2}}} $$
$$ y = \frac{\sqrt{\log x}}{e^{\frac{x^2}{2}}} $$
More Information
The simplified expression represents the original equation in a more explicit form, showing the square root of the logarithm in the numerator and an exponential function in the denominator.
Tips
A common mistake might be incorrectly applying the power rule to the exponential function or misinterpreting the square root as applying only to the logarithm.
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