Question

The value of the expression \( \frac{81^3 + 28^3 - 109^3}{86^2 - 23^2} \) is?

Hint:

When you see expressions with cubes or squares of numbers that seem related, always look for an underlying algebraic identity. Checking if the sum of the bases in a sum of cubes expression is zero is a very common shortcut.

Answer 1

Step 1: Understanding the Concept:
This problem can be simplified significantly by using algebraic identities. The numerator relates to the identity involving \(a^3+b^3+c^3\), and the denominator is a direct application of the difference of squares identity.
Step 2: Key Formula or Approach:
We will use the following identities:

Difference of Squares: \(a^2 - b^2 = (a - b)(a + b)\)
Sum of Cubes variant: If \(a + b + c = 0\), then \(a^3 + b^3 + c^3 = 3abc\). This can be rearranged from the general identity \(a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)\).
Step 3: Detailed Explanation:
Simplifying the Numerator:
Let \(a = 81\), \(b = 28\), and \(c = -109\).
Let's check the sum \(a + b + c\). \[ a + b + c = 81 + 28 + (-109) = 109 - 109 = 0 \] Since \(a+b+c=0\), we can use the identity \(a^3 + b^3 + c^3 = 3abc\).
The numerator is \(81^3 + 28^3 + (-109)^3\). \[ \text{Numerator} = 3 \times (81) \times (28) \times (-109) = -3 \times 81 \times 28 \times 109 \] Simplifying the Denominator:
The denominator is \(86^2 - 23^2\), which is in the form \(a^2 - b^2\). Using the identity \(a^2 - b^2 = (a - b)(a + b)\): \[ \text{Denominator} = (86 - 23)(86 + 23) = (63)(109) \] Evaluating the Expression:
Now, we put the simplified numerator and denominator together. \[ \text{Expression} = \frac{-3 \times 81 \times 28 \times 109}{63 \times 109} \] We can cancel the 109 from the numerator and denominator. \[ \text{Expression} = \frac{-3 \times 81 \times 28}{63} \] We know that \(63 = 9 \times 7\). \[ \text{Expression} = \frac{-3 \times 81 \times 28}{9 \times 7} \] Now, simplify the fraction: \[ \text{Expression} = -3 \times \left(\frac{81}{9}\right) \times \left(\frac{28}{7}\right) \] \[ \text{Expression} = -3 \times 9 \times 4 \] \[ \text{Expression} = -108 \] Step 4: Final Answer:
The value of the expression is -108.