3. Take the principal square root of both sides.
[tex]\[
\sqrt{2a^2} = \sqrt{c^2}
\][/tex]

4. Simplify:
[tex]\[
\sqrt{2}a = c
\][/tex]



Answer :

To solve the equation, let’s carefully follow the steps provided and simplify it at each stage.

Given equation:

[tex]\[ \sqrt{2a^2} = \sqrt{c^2} \][/tex]

### Step 3: Taking the Principal Square Root of Both Sides

We start with the equation [tex]\( \sqrt{2a^2} = \sqrt{c^2} \)[/tex]. Here, we are taking the square root of both expressions:

[tex]\[ \sqrt{2a^2} \][/tex]
[tex]\[ \sqrt{c^2} \][/tex]

### Step 4: Simplification

Now, let's simplify both sides separately.

1. Simplifying the left side:

[tex]\[ \sqrt{2a^2} = \sqrt{2} \cdot \sqrt{a^2} \][/tex]

The square root of a product is the product of the square roots. Recognizing that, we can break down the left side:

[tex]\[ \sqrt{a^2} = |a| \][/tex]

So,

[tex]\[ \sqrt{2} \cdot \sqrt{a^2} = \sqrt{2} \cdot |a| \][/tex]

Which simplifies to:

[tex]\[ \sqrt{2} \cdot |a| \][/tex]

2. Simplifying the right side:

[tex]\[ \sqrt{c^2} \][/tex]

The square root of [tex]\( c^2 \)[/tex] is the absolute value of [tex]\( c \)[/tex]:

[tex]\[ \sqrt{c^2} = |c| \][/tex]

Therefore, we can rewrite the original equation as:

[tex]\[ \sqrt{2} \cdot |a| = |c| \][/tex]

Thus, the simplified form of the given equation is:

[tex]\[ \sqrt{2} \cdot |a| = |c| \][/tex]

In summary, we have the simplified result:

[tex]\[ \sqrt{2} \cdot |a| \text{ equals } |c| \][/tex]

This is your simplified equation and the answer to the problem.