91Ó°ÊÓ

When confronted with an equation, one common problem is solving for a specific variable in that equation. This simply means figuring out what value the variable represents. Let's break down the process using the provided equation, \(E = \frac{I}{d^2}\). We want to solve for the variable \(d\). Here, 'solving for' a variable means rearranging the equation so that the variable of interest is on its own on one side of the equation, with everything else on the other side.

• We start by identifying the variable we're interested in, which is \(d\) in this case.
• The goal is to manipulate the equation to make \(d\) the subject.
• This often requires performing operations such as multiplication, division, or applying roots to both sides of the equation.

Initially, \(d\) is part of a fraction in our given equation. Thus, we need to perform operations that help free \(d\) from being in the denominator. Keeping track of each step is crucial to ensure accuracy and efficiency.

Isolating a variable means rearranging an equation to get the variable by itself. This is often achieved through inverse operations—essentially doing the opposite of what is currently being done to the variable of interest.

• In our equation \(E = \frac{I}{d^2}\), the first step is to remove \(d\) from the denominator by multiplying both sides of the equation by \(d^2\). This gives \(d^2E = I\).
• Next, we continue isolating \(d^2\) by dividing both sides by \(E\), resulting in \(d^2 = \frac{I}{E}\).

Once you isolate \(d^2\), you focus on finding \(d\) instead, which leads to applying another math operation, the square root, to both sides. Isolating \(d\) involves thinking strategically about which operations will simplify the equation efficiently and correctly. Be patient, and apply these steps carefully.

The square root operation is essential when solving for a variable that is squared in an equation. In mathematical terms, taking the square root is the inverse of squaring. In the equation we are working with, after isolating \(d^2\), we arrive at \(d^2 = \frac{I}{E}\); to find \(d\), we take the square root of both sides.

• Taking the square root means finding a value that when multiplied by itself gives the original number. For \(d^2 = x\), \(d = \sqrt{x}\).
• Thus, with \(d^2 = \frac{I}{E}\), applying the square root to both sides yields \(d = \sqrt{\frac{I}{E}}\).

This approach efficiently finalizes the isolation of \(d\) and provides the solution for \(d\). Remember, taking the square root can lead to two possible answers (positive and negative roots) when dealing with real numbers, unless specified otherwise by the context. Here, the positive root is often sought, especially in physical contexts where dimensions make sense as positive numbers.