The given equation is:

\[
\sqrt{(x-70)^2} = -6x
\]

### Step 1: Simplify the square root expression
The square root of a square simplifies as follows:

\[
\sqrt{(x-70)^2} = |x - 70|
\]

So, the equation becomes:

\[
|x - 70| = -6x
\]

### Step 2: Analyze the absolute value equation
Since the absolute value of a number is always non-negative, the left side, \(|x - 70|\), is always greater than or equal to 0. However, the right side, \(-6x\), is negative or zero depending on the value of \(x\).

This implies that the equation \(|x - 70| = -6x\) can only hold true if \(-6x \geq 0\), which means \(x \leq 0\).

### Step 3: Solve for \(x\)
Now, we need to consider two cases for the absolute value equation.

#### Case 1: \(x - 70 = -6x\)

Solving this:

\[
x - 70 = -6x
\]
\[
x + 6x = 70
\]
\[
7x = 70
\]
\[
x = 10
\]

However, for this solution, we need to check if it satisfies the condition \(x \leq 0\). Since \(x = 10\) does not satisfy \(x \leq 0\), it is not a valid solution.

#### Case 2: \(-(x - 70) = -6x\), or \(70 - x = -6x\)

Solving this:

\[
70 - x = -6x
\]
\[
70 = -5x
\]
\[
x = -\frac{70}{5} = -14
\]

Since \(x = -14\) satisfies the condition \(x \leq 0\), this is a valid solution.

### Conclusion:
The solution to the equation is:

\[
x = -14
\]

Would you like more details, or do you have any questions?

Here are five follow-up questions for further exploration:
1. How would the solution change if the right side of the equation were \(6x\) instead of \(-6x\)?
2. What happens if the absolute value involves a different linear expression, like \(|x + 5|\)?
3. Can you solve an equation with a cube root instead of a square root, such as \(\sqrt[3]{(x-70)^3} = -6x\)?
4. How do you handle absolute value inequalities, such as \(|x - 70| > -6x\)?
5. What are the implications of an equation where both sides are negative or positive absolute values?

**Tip**: When solving absolute value equations, always check the conditions for both cases of the absolute value and ensure the solutions meet the original constraints.

Learn how to solve the absolute value equation √{(x-70)^2} = -6x step-by-step. This algebraic problem explores the properties of square roots, absolute values, and solving linear equations. Suitable for students in Grades 9-12.

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