Question

Find the value of the following algebraic expression: \[ \frac{(0.64)^{3}-(0.36)^{3}}{(0.64)^{2}+(0.64)(0.36)+(0.36)^{2}} = \]

• A. ( 725 )
• B. ( 1125 )
• C. ( 750 )
• D. ( 1150 )

Hint:

Whenever you see a structure that resembles (a^3 - b^3a^2 + ab + b^2), do not waste valuable exam time computing squares or cubes! The entire fraction immediately simplifies down to just ((a - b)). If it is a sum of cubes in the format (a^3 + b^3a^2 - ab + b^2), the result is simply ((a + b)).

Answer 1

The Correct Option is A

Concept: This question can be simplified by identifying the standard algebraic identity for the difference of two cubes: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \] By rewriting this identity, the ratio of the difference of cubes to its quadratic factor can be simplified directly: \[ \frac{a^3 - b^3}{a^2 + ab + b^2} = a - b \]

Step 1: Let us substitute variables for the decimal numbers to track the simplification easily. Let: \[ a = 0.64 \quad \text{and} \quad b = 0.36 \]

Step 2: Substitute these variable definitions into the given algebraic expression: \[ \text{Expression} = \frac{a^3 - b^3}{a^2 + ab + b^2} \]

Step 3: Apply the algebraic identity expansion (a^3 - b^3 = (a - b)(a^2 + ab + b^2)) to the numerator: \[ \text{Expression} = \frac{(a - b)(a^2 + ab + b^2)}{a^2 + ab + b^2} \] Since the term ((a^2 + ab + b^2)) appears identically in both the numerator and the denominator, they cancel each other out completely: \[ \text{Expression} = a - b \]

Step 4: Substitute the actual numeric decimal values back into the expression: \[ \text{Expression} = 0.64 - 0.36 = 0.28 \]

Step 5: Convert the resulting decimal number into its fractional form: \[ 0.28 = \frac{28}{100} \] Now, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4: \[ \frac{28 \div 4}{100 \div 4} = \frac{7}{25} \] Thus, the value matches Option (A) exactly.