Correct versus Incorrect

Correct:
$$\sqrt {x^2}= \left| x \right|$$

Incorrect:
$(x^2)^{1/2}=x$

Correct:
$$(x^2)^{1/2}=\left| x \right|$$

Also Correct:
$(x^2)^{1/2}=x$ for $x>0$

In general, Correct:
$(a^b)^c=a^{(bc)}$ for $a>0$


Example Incorrect:
$\sqrt {\sin^2\theta \cos^2 \theta}=\sin \theta \cos \theta$

Example Correct:
$\sqrt {\sin^2\theta \cos^2 \theta}=\left| \sin \theta \cos \theta \right|$


Incorrect:
$e^{(a+b)}=e^a + e^b$

Correct:
$$e^{(a+b)}=e^a e^b$$


Incorrect:
$x=t^2$ implies $t=\sqrt{x}$

Correct:
$x=t^2$ implies $t=\pm \sqrt{x}$


Correct:
$$\int \frac{1}{x}dx = \ln \left| x\right|+C$$


Incorrect:
$t(3t-2)=t$ implies $3t-2=1$ and therefore $t=1$.

Correct:
$t(3t-2)=t$ implies $t(3t-2)-t=0$ implies $t(3t-3)=0$ implies $t=0$ or $3t-3=0$ implies $t=0$ or $t=1$.

Also Correct:
$t(3t-2)=t$. Case 1: $t\neq 0$. Then $3t-2=1$ and therefore $t=1$. Case 2: $t=0$. We check and see this is a solution. We conclude $t=0$ or $t=1$.


Incorrect:
$\frac{1}{a+b}=\frac{1}{a}+\frac{1}{b}$


Incorrect:
$\ln(x)y=e^x+C$ implies $y=\frac{e^x}{\ln(x)}+C$

Correct:
$\ln(x)y=e^x+C$ implies $y=\frac{e^x+C}{\ln{x}}$


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