Evaluating Algebraic Expressions with Given Values

The problem asks us to find the value of the expression \({{a^3 \ - \ b^{3}} \over a^2 \ - \ b^{2}}\) - \({{3ab} \over a \ + \ b}\) when \(a = 26\) and \(b = 22\).

Let's simplify the given expression first before substituting the values of \(a\) and \(b\). The expression is:

\[ \text{Expression} = \frac{a^3 - b^3}{a^2 - b^2} - \frac{3ab}{a + b} \]

We can use the algebraic identities for the difference of cubes and the difference of squares:

• Difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
• Difference of squares: \(a^2 - b^2 = (a - b)(a + b)\)

Using these identities, the first term of the expression can be simplified:

\[ \frac{a^3 - b^3}{a^2 - b^2} = \frac{(a - b)(a^2 + ab + b^2)}{(a - b)(a + b)} \]

Assuming \(a \neq b\), we can cancel the \((a - b)\) term from the numerator and denominator:

\[ \frac{a^3 - b^3}{a^2 - b^2} = \frac{a^2 + ab + b^2}{a + b} \]

Now substitute this back into the original expression:

\[ \text{Expression} = \frac{a^2 + ab + b^2}{a + b} - \frac{3ab}{a + b} \]

Since both terms have the same denominator \((a + b)\), we can combine the numerators:

\[ \text{Expression} = \frac{(a^2 + ab + b^2) - (3ab)}{a + b} \]

\[ \text{Expression} = \frac{a^2 + ab + b^2 - 3ab}{a + b} \]

Combine the like terms in the numerator (\(ab - 3ab = -2ab\)):

\[ \text{Expression} = \frac{a^2 - 2ab + b^2}{a + b} \]

Recognize that the numerator \(a^2 - 2ab + b^2\) is the algebraic identity for the perfect square of a difference:

• Perfect square of a difference: \(a^2 - 2ab + b^2 = (a - b)^2\)

So, the simplified expression is:

\[ \text{Expression} = \frac{(a - b)^2}{a + b} \]

Now, substitute the given values \(a = 26\) and \(b = 22\) into the simplified expression:

\[ \text{Value} = \frac{(26 - 22)^2}{26 + 22} \]

First, calculate the values inside the parentheses:

• \(26 - 22 = 4\)
• \(26 + 22 = 48\)

Substitute these values back into the expression:

\[ \text{Value} = \frac{(4)^2}{48} \]

Calculate the square in the numerator:

\[ \text{Value} = \frac{16}{48} \]

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 16:

\[ \text{Value} = \frac{16 \div 16}{48 \div 16} = \frac{1}{3} \]

Thus, the value of the expression is \({{1} \over 3}\).

Revision Table: Key Steps for Evaluating Expressions

Additional Information: Useful Algebraic Identities

Simplifying expressions often requires knowledge of common algebraic identities. Here are some key identities used in this problem and related ones:

• Difference of Squares: \(a^2 - b^2 = (a - b)(a + b)\)
• Sum of Squares: \(a^2 + b^2\) (does not factor into real linear terms)
• Perfect Square Trinomials:
  • \((a + b)^2 = a^2 + 2ab + b^2\)
  • \((a - b)^2 = a^2 - 2ab + b^2\)
• Difference of Cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
• Sum of Cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)

Knowing these identities helps in quickly simplifying complex algebraic fractions and expressions, which is crucial for efficiently solving problems like the one presented.