Evaluating the Mathematical Expression

The problem asks for the value of a specific mathematical expression. The expression involves sums and differences of two numbers, raised to the power of two, and then divided by the sum of the squares of the same numbers.

The given expression is:

\(\frac{{{{\left( {443{\rm{\;}} + {\rm{\;}}547} \right)}^2} + {\rm{\;}}{{\left( {443{\rm{\;}} - {\rm{\;}}547} \right)}^2}{\rm{\;}}}}{{443{\rm{\;}} \times {\rm{\;}}443{\rm{\;}} + {\rm{\;}}547{\rm{\;}} \times {\rm{\;}}547}}\)

Using Algebraic Identities for Simplification

We can simplify this expression by using a standard algebraic identity. Let's represent the two numbers, 443 and 547, by variables \(a\) and \(b\).

• Let \(a = 443\)
• Let \(b = 547\)

Substituting \(a\) and \(b\) into the expression, we get:

\(\frac{{{{\left( {a + b} \right)}^2} + {\rm{\;}}{{\left( {a - b} \right)}^2}{\rm{\;}}}}{{{a^2}{\rm{\;}} + {\rm{\;}}{b^2}}}\)

Now, recall the algebraic identities for \((a+b)^2\) and \((a-b)^2\):

• \({(a + b)^2} = a^2 + 2ab + b^2\)
• \({(a - b)^2} = a^2 - 2ab + b^2\)

Let's substitute these expanded forms into the numerator of our expression:

Numerator \( = {(a + b)^2} + {(a - b)^2} = (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2)\)

Combine like terms in the numerator:

Numerator \( = a^2 + a^2 + 2ab - 2ab + b^2 + b^2\)

Numerator \( = 2a^2 + 0 + 2b^2\)

Numerator \( = 2a^2 + 2b^2\)

We can factor out a 2 from the numerator:

Numerator \( = 2(a^2 + b^2)\)

Now, let's look at the denominator of the original expression:

Denominator \( = 443{\rm{\;}} \times {\rm{\;}}443{\rm{\;}} + {\rm{\;}}547{\rm{\;}} \times {\rm{\;}}547\)

Using our substitution \(a=443\) and \(b=547\), the denominator is:

Denominator \( = a^2 + b^2\)

So, the entire expression becomes:

\(\frac{{2(a^2 + b^2)}}{{a^2 + b^2}}\)

Calculating the Final Value

Assuming that \(a^2 + b^2 \neq 0\) (which is true since 443 and 547 are non-zero numbers, so their squares are positive, and their sum is positive), we can cancel the term \((a^2 + b^2)\) from both the numerator and the denominator.

\(\frac{{2\cancel{(a^2 + b^2)}}}{{\cancel{(a^2 + b^2)}}} = 2\)

Thus, the value of the given expression is 2.

Steps to Evaluate the Expression

1. Identify the structure of the expression and notice the pattern involving the sum and difference of two numbers squared.
2. Assign variables \(a\) and \(b\) to the numbers (443 and 547).
3. Rewrite the expression using \(a\) and \(b\).
4. Apply the algebraic identity \((a+b)^2 + (a-b)^2 = 2(a^2 + b^2)\) to simplify the numerator.
5. Identify the denominator as \(a^2 + b^2\).
6. Substitute the simplified numerator back into the expression.
7. Cancel the common term \((a^2 + b^2)\) from the numerator and denominator.
8. The result after cancellation is the value of the expression.

Revision Table: Key Concepts Revisited

Additional Information: Related Identities

Understanding algebraic identities is key to simplifying complex expressions quickly. Here are a few more related identities:

• Difference of Squares: \(a^2 - b^2 = (a+b)(a-b)\)
• Perfect Square Trinomials:
  \((a+b)^2 = a^2 + 2ab + b^2\)
  \((a-b)^2 = a^2 - 2ab + b^2\)
• Sum of Cubes: \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)
• Difference of Cubes: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)
• Expansion of \((a+b+c)^2\): \((a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\)

These identities help solve various problems involving factorization and simplification of polynomials and rational expressions.