Simplify square root of (3-1)^2+(3-1)^2

Solution:

Step 1:

Compute $3-1$. $\sqrt{(3-1{)}^2+(3-1{)}^2}$

Step 2:

Calculate the square of $2$. $\sqrt{2^2+(3-1{)}^2}$

Step 3:

Again, compute $3-1$. $\sqrt{2^2+2^2}$

Step 4:

Calculate the square of $2$. $\sqrt{4+4}$

Step 5:

Combine the sum of $4$ and $4$. $\sqrt{8}$

Step 6:

Express $8$ as $2^2\cdot 2$.

Step 6.1:

Extract $4$ from $8$. $\sqrt{4\cdot 2}$

Step 6.2:

Represent $4$ as $2^2$. $\sqrt{2^2\cdot 2}$

Step 7:

Extract the square root of the perfect square. $2\sqrt{2}$

Step 8:

Present the final result in various forms.

Exact Form: $2\sqrt{2}$ Decimal Form: Approximately $2.82842712$

Knowledge Notes:

The problem involves simplifying a square root expression that contains squared terms. The steps taken to simplify the expression are based on the following knowledge points:

1. Basic Arithmetic Operations: Subtraction and addition are used to simplify the expression inside the square root.

2. Exponentiation: Squaring a number means multiplying it by itself (e.g., $2^2=2\times 2=4$).

3. Square Root Properties: The square root of a square number is the base of the square (e.g., $\sqrt{2^2}=2$). Also, the square root of a product of a square number and another number allows the square number to be taken out of the square root (e.g., $\sqrt{2^2\cdot 2}=2\sqrt{2}$).

4. Simplification of Radicals: When simplifying square roots, any factor that is a perfect square can be taken out of the radical, reducing the expression to its simplest form.

5. Decimal Approximation: Exact expressions involving square roots can be approximated to decimal form using a calculator or other computational tools.