# Equations with Exponents

So let see how to solve ${x}^{2}\hspace{0.33em}{-}\hspace{0.33em}{7}\hspace{0.33em}{=}\hspace{0.33em}{18}$. A good first step would be to remove the -7 from the left side. We do this by adding +7:

${x}^{2}\hspace{0.33em}{-}\hspace{0.33em}{7}\hspace{0.33em}{=}\hspace{0.33em}{18}\hspace{0.33em}\hspace{0.33em}\Longrightarrow\hspace{0.33em}\hspace{0.33em}{x}^{2}\hspace{0.33em}{-}\hspace{0.33em}{7}\hspace{0.33em}{+}\hspace{0.33em}{7}\hspace{0.33em}{=}\hspace{0.33em}{18}\hspace{0.33em}{+}\hspace{0.33em}{7}\hspace{0.33em}\hspace{0.33em}\Longrightarrow\hspace{0.33em}{x}^{2}\hspace{0.33em}{=}\hspace{0.33em}{25}$

Now before I proceed further, after doing things like adding or subtracting a number to both sides of an equation, you start to notice mental shortcuts. For example, adding the 7 to both sides of the equation, effectively is the same as moving the 7 from the left side to the right side and changing the sign. This can be done with any term: move it to the other side of the equation and change the sign. By the way, a term is anything that is added or subtracted to or from the entire side of an equation. I’ll illustrate this further in future equations.

So now we have ${x}^{2}\hspace{0.33em}{=}\hspace{0.33em}{25}$. So how do we get x by itself? From yesterday’s post, we know that:

$\sqrt{{x}^{2}}\hspace{0.33em}{=}\hspace{0.33em}{x}$

So this suggests that we need to take the square root of both sides of the equation:

${x}^{2}\hspace{0.33em}{=}\hspace{0.33em}{25}\hspace{0.33em}\hspace{0.33em}\Longrightarrow\hspace{0.33em}\hspace{0.33em}\sqrt{{x}^{2}}\hspace{0.33em}{=}\hspace{0.33em}\sqrt[]{25}\hspace{0.33em}\hspace{0.33em}\Longrightarrow\hspace{0.33em}\hspace{0.33em}{x}\hspace{0.33em}{=}\hspace{0.33em}\pm\hspace{0.33em}{5}$

Did you remember that taking the square root results in two solutions? If you replace the x in the original equation with either +5 or -5, you will get a true equation 18 = 18.

Now many time the equation comes from a physical problem, like solving for a length. If so, then you can ignore the negative solution as that would not make sense. There are times, however, when the negative solution is the desired one.