Simplify square root of (5-3)2+(5-3)2

Question

Solvely Expert · Accepted Answer

Compute the difference between $5$ and $3$ . $\sqrt{(5 - 3)^{2} + (5 - 3)^{2}}$

Calculate the square of $2$ . $\sqrt{2^{2} + (5 - 3)^{2}}$

Again, compute the difference between $5$ and $3$ . $\sqrt{2^{2} + (5 - 3)^{2}}$

Calculate the square of $2$ once more. $\sqrt{2^{2} + 2^{2}}$

Combine the results of the squares. $\sqrt{4 + 4}$

Express the sum under the radical as a product of prime factors.

Factor out the square of $2$ from $8$ . $\sqrt{4 \cdot 2}$

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

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

Present the final answer in various forms.

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

To solve the given problem, we apply the following knowledge points:

Square of a Difference: The square of a difference formula is $(a - b)^{2} = a^{2} - 2 a b + b^{2}$ . 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.

Simplify square root of (5-3)^2+(5-3)^2