Certainly! Let's solve the given literal equation for [tex]\( m \)[/tex] step-by-step.
We start with the equation:
[tex]\[ t = 2\pi \sqrt{\frac{m}{k}} \][/tex]
Our goal is to solve for [tex]\( m \)[/tex].
### Step 1: Isolate the Square Root Term
We need to get rid of the square root. First, we need to isolate the term that contains the square root:
[tex]\[ t = 2\pi \sqrt{\frac{m}{k}} \][/tex]
### Step 2: Square Both Sides
To remove the square root, we square both sides of the equation:
[tex]\[ t^2 = \left(2\pi \sqrt{\frac{m}{k}}\right)^2 \][/tex]
Simplifying the right-hand side, we use the property [tex]\((a \cdot b)^2 = a^2 \cdot b^2\)[/tex]:
[tex]\[ t^2 = (2\pi)^2 \left(\sqrt{\frac{m}{k}}\right)^2 \][/tex]
Since [tex]\((2\pi)^2 = 4\pi^2\)[/tex] and [tex]\(\left(\sqrt{\frac{m}{k}}\right)^2 = \frac{m}{k}\)[/tex], this simplifies to:
[tex]\[ t^2 = 4\pi^2 \frac{m}{k} \][/tex]
### Step 3: Isolate [tex]\( m \)[/tex]
To isolate [tex]\( m \)[/tex], we need to get rid of the fraction. We'll multiply both sides of the equation by [tex]\( k \)[/tex]:
[tex]\[ k t^2 = 4\pi^2 \cdot m \][/tex]
### Step 4: Solve for [tex]\( m \)[/tex]
Finally, to solve for [tex]\( m \)[/tex], we divide both sides by [tex]\( 4\pi^2 \)[/tex]:
[tex]\[ m = \frac{k t^2}{4\pi^2} \][/tex]
So the solution to the equation, solving for [tex]\( m \)[/tex], is:
[tex]\[ \boxed{m = \frac{k t^2}{4\pi^2}} \][/tex]
