The given equation is \( (C - 7)^2 = 81 \).

To solve for \( C \), follow these steps:

### Step 1: Take the square root of both sides
Take the square root of both sides of the equation to eliminate the square:

\[
\sqrt{(C - 7)^2} = \pm \sqrt{81}
\]

This simplifies to:

\[
C - 7 = \pm 9
\]

### Step 2: Solve for \( C \)
Now, solve the equation for \( C \) by considering the two possible cases:

1. \( C - 7 = 9 \)
2. \( C - 7 = -9 \)

#### Case 1: \( C - 7 = 9 \)
Add 7 to both sides:

\[
C = 9 + 7 = 16
\]

#### Case 2: \( C - 7 = -9 \)
Add 7 to both sides:

\[
C = -9 + 7 = -2
\]

### Final Answer
The solutions for \( C \) are:

\[
C = 16 \quad \text{or} \quad C = -2
\]

Would you like more details or have any questions?

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Here are some related questions you might be interested in:

1. How do you solve equations involving absolute values, like \( |C - 7| = 9 \)?
2. What is the process for solving quadratic equations like \( (C - 3)^2 = 49 \)?
3. How do you solve equations where both sides are squared, such as \( (2C + 1)^2 = (C - 3)^2 \)?
4. What steps would you take to solve \( (C + 5)^2 = 36 \)?
5. How do you solve quadratic equations using the method of completing the square?
6. What is the significance of the ± symbol in solutions involving square roots?
7. How do you solve quadratic equations using the quadratic formula?
8. What happens if the equation involves a perfect square trinomial?

**Tip:** Always remember to consider both the positive and negative square roots when solving equations like \( (x - a)^2 = b \).

Learn how to solve the equation (c-7)^2 = 81 step-by-step with detailed explanations and solutions. This problem covers key algebraic concepts such as quadratic equations and the square root property theorem. Suitable for students in Grades 7-9.

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