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The technique known as 'extraction of roots' is a fundamental approach used to solve quadratic equations that are in a specific form. This form is characterized by having the variable squared and set equal to a numerical value.

In our exercise, the equation is given as \( c^2 = 81 \). To extract the roots, we essentially 'undo' the squaring operation to find what number, when squared, gives us 81. The mathematical operation that reverses squaring is the square root.

It's important to recall that squaring any real number always gives a positive result. Therefore, when we take the square root of both sides, we must consider both the positive and negative square root because both \( 9^2 \) and \( (-9)^2 \) yield 81. This is why we write \( c = \text{\pm\sqrt{81}} \) as part of the solution process.

When we talk about square roots in the context of solving quadratic equations, we're focusing on the inverse operation of squaring a number. The square root of any non-negative number 'a' is a number 'b' such that \( b^2 = a \). It's represented by the radical symbol \( \sqrt{ } \).

In practice, taking the square root can be thought of as asking, 'What number multiplied by itself gives me this particular value?' For example, since \( 9 \times 9 = 81 \), we can say that 81 has a square root of 9, written as \( \sqrt{81} = 9 \).

It is critical to remember that any positive real number has two square roots: One is positive, and the other is negative. However, zero has only one square root, which is itself (0). When we apply this to the equation in our exercise, we consider both \( +9 \) and \( -9 \) as square roots of 81.

To 'isolate the variable' means to rearrange an algebraic equation so that the variable we are solving for appears by itself on one side of the equation, with all other terms on the opposite side. This process is the crux of solving most algebra problems and is crucial in solving quadratic equations by extraction of roots.

In our example, \( c^2 = 81 \), the variable \( c \) is not isolated because it is squared. To isolate \( c \), we perform the inverse operation of squaring, which is finding the square root. When we apply the square root to both sides of the equation \( \sqrt{c^2} = \text{\pm\sqrt{81}} \), we are left with \( c = \text{\pm\sqrt{81}} \). This step perfectly isolates \( c \) and provides us with the solutions once the square root of 81 is calculated. As such, the simple act of isolating the variable forms a pathway to finding the solutions for \( c \).