Simplify square root of (5-3)^2+(5-3)^2
The question is asking for the simplification of an arithmetic expression involving square roots and squares. Specifically, it requests to take the square root of a sum where each term in the sum is a square of an arithmetic difference (5-3). The expression contains the same difference squared twice, added together, and the question seeks the simplified form after these operations are carried out.
$\sqrt{\left(\left(\right. 5 - 3 \left.\right)\right)^{2} + \left(\left(\right. 5 - 3 \left.\right)\right)^{2}}$
Answer
Solution:
Step 1:
Compute the difference between $5$ and $3$. $\sqrt{(5-3)^2 + (5-3)^2}$
Step 2:
Calculate the square of $2$. $\sqrt{2^2 + (5-3)^2}$
Step 3:
Again, compute the difference between $5$ and $3$. $\sqrt{2^2 + (5-3)^2}$
Step 4:
Calculate the square of $2$ once more. $\sqrt{2^2 + 2^2}$
Step 5:
Combine the results of the squares. $\sqrt{4 + 4}$
Step 6:
Express the sum under the radical as a product of prime factors.
Step 6.1:
Factor out the square of $2$ from $8$. $\sqrt{4 \cdot 2}$
Step 6.2:
Represent $4$ as the square of $2$. $\sqrt{2^2 \cdot 2}$
Step 7:
Extract the square root of the perfect square outside the radical. $2\sqrt{2}$
Step 8:
Present the final answer in various forms.
Exact Form: $2\sqrt{2}$ Decimal Form: Approximately $2.82842712$
Knowledge Notes:
To solve the given problem, we apply the following knowledge points:
Square of a Difference: The square of a difference formula is $(a - b)^2 = a^2 - 2ab + b^2$. In this case, $(5-3)^2$ simplifies directly to $2^2$ because $5-3$ equals $2$.
Exponentiation: Raising a number to the power of $2$ means multiplying the number by itself. For example, $2^2 = 2 \times 2 = 4$.
Simplifying Square Roots: The square root of a sum $\sqrt{a^2 + b^2}$ cannot be simplified directly unless $a$ and $b$ have common factors that are perfect squares.
Factoring Perfect Squares: A number like $8$ can be factored into $2^2 \cdot 2$, which helps in simplifying the square root.
Radical Simplification: When a term under a radical is a perfect square, it can be taken out of the radical as its square root. For instance, $\sqrt{2^2 \cdot 2}$ simplifies to $2\sqrt{2}$.
Approximation: The square root of a non-perfect square, such as $\sqrt{2}$, can be approximated using a calculator. The decimal approximation of $\sqrt{2}$ is about $1.41421356$, and when multiplied by $2$, it gives approximately $2.82842712$.
By following these steps and applying these knowledge points, we can simplify the given expression to its exact and decimal forms.
