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.
