Simplifying a Mathematical Expression using Algebraic Identity

The given mathematical expression is:

\(\dfrac{(5.17-2.19)^2-(5.17+2.19)^2}{11.3223}\)

Let's denote \(a = 5.17\) and \(b = 2.19\). The expression can be written in the form:

\(\dfrac{(a-b)^2 - (a+b)^2}{c}\)

where \(c = 11.3223\).

Step-by-Step Simplification

We can simplify the numerator using algebraic identities. Recall the identities:

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

Now, let's subtract \((a+b)^2\) from \((a-b)^2\):

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

Distribute the minus sign:

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

Group like terms:

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

\(= 0 + 0 - 4ab\)

\(= -4ab\)

Substituting Values into the Expression

Now substitute the simplified numerator back into the original expression:

\(\dfrac{-4ab}{c}\)

Substitute the values \(a = 5.17\), \(b = 2.19\), and \(c = 11.3223\):

\(\dfrac{-4 \times 5.17 \times 2.19}{11.3223}\)

Calculating the Product of a and b

Let's calculate the product \(a \times b = 5.17 \times 2.19\):

5.17

x 2.19

------

4653 (5.17 x 0.09)

5170 (5.17 x 0.10)

103400 (5.17 x 2.00)

------

11.3223 (Summing the results, keeping track of decimal places)

So, \(5.17 \times 2.19 = 11.3223\).

Completing the Calculation

Now substitute the product \(ab\) back into the expression:

\(\dfrac{-4 \times 11.3223}{11.3223}\)

The term \(11.3223\) appears in both the numerator and the denominator. Assuming \(11.3223 \neq 0\), we can cancel it out:

\(= -4\)

Conclusion

The value of the expression \(\dfrac{(5.17-2.19)^2-(5.17+2.19)^2}{11.3223}\) is \(-4\).

Revision Table: Algebraic Identities

Additional Information: Simplifying Expressions

Simplifying mathematical expressions involves rewriting them in a simpler, more understandable form. This often uses algebraic identities, factorization, combining like terms, and applying the order of operations (PEMDAS/BODMAS).

• Why simplify? Simplifying makes calculations easier, helps in solving equations, and reveals the structure of an expression.
• Key techniques:
  • Use distributive property: \(a(b+c) = ab + ac\)
  • Combine like terms: \(3x + 2y - x = 2x + 2y\)
  • Apply algebraic identities: As used in this problem.
  • Factorize expressions: Find common factors or use identities in reverse.
• Numerator and Denominator: When dealing with fractions, simplify the numerator and the denominator separately before dividing, or look for common factors to cancel out, as we did here.
• Special Cases: Be mindful of division by zero. In this problem, the denominator \(11.3223\) is not zero, so the division is valid.

Understanding algebraic identities is a fundamental skill for simplifying complex expressions efficiently.