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

Step 1: Spot the hidden identity.
The expression looks long, but it follows the difference of cubes pattern, $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$. The bottom of the fraction is exactly that second bracket.

Step 2: Name the two numbers.
Let $a = 0.64$ and $b = 0.36$. Then the top is $a^3 - b^3$ and the bottom is $a^2 + ab + b^2$.

Step 3: Cancel using the identity.
Replace the top with its factored form. \[ \frac{(a-b)(a^2+ab+b^2)}{a^2+ab+b^2} \] The big bracket appears on top and bottom, so it cancels neatly.

Step 4: What is left.
After cancelling, only $a - b$ remains. This is the whole point of the identity, it turns a heavy calculation into a simple subtraction.

Step 5: Subtract the decimals.
\[ a - b = 0.64 - 0.36 = 0.28 \]

Step 6: Match to the option.
The value $0.28$ is what the listed option represents. So the expression equals \[ \boxed{0.28} \]