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Chapter 8: Problem 63

Solve formula for the indicated variable. \(z=\sqrt{\frac{L}{C}}\) for \(L\)

Short Answer

Expert verified
\[L = z^2 \cdot C\]

Step by step solution

01

Isolate the Square Root

The formula given is \[z = \sqrt{\frac{L}{C}}\]. To solve for the variable L, first eliminate the square root by squaring both sides of the equation. This will give us: \[z^2 = \frac{L}{C}\].
02

Solve for L

Now, to isolate L, multiply both sides of the equation by C: \[L = z^2 \cdot C\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating the Variable
Isolating the variable is an essential first step in solving equations. The goal here is to get the variable you are solving for all by itself, or isolated, on one side of the equation.
Let's start with the formula given: \( z = \sqrt{\frac{L}{C}} \)
To solve for \ L \ , we first need to make sure that \ L \ is isolated.
This can be done by removing anything else that is around \ L \ .
In our case, the square root is the first obstacle to isolating \ L \ .
Square Root Elimination
Square root elimination is one way to simplify the equation so that the variable is easier to isolate.
In the equation \( z = \sqrt{\frac{L}{C}} \), the square root is keeping \ L \ from being isolated.
To eliminate the square root, we square both sides of the equation.
This is an algebraic rule where if two values are equal, their squares are also equal.
So, we get:
\( z^2 = \frac{L}{C} \)
This step has removed the square root and made the equation simpler to solve.
Algebraic Manipulation
Algebraic manipulation involves rearranging the given equation using basic algebraic operations like addition, subtraction, multiplication, and division.
Our goal now is to solve for \ L \ in the equation \( z^2 = \frac{L}{C} \).
To isolate \ L \, we need to get rid of \ \frac{1}{C} \ .
We do this by multiplying both sides of the equation by \ C \:
\( L = z^2 \cdot C \)
And there we have it! We have isolated \ L \ and solved the equation. This final step is crucial, as algebraic manipulation helps simplify the equation, making it straightforward to find the required variable.

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