Question

The value of $ \frac{(2.3)^3 - 0.027{(2.3)^2 + 0.69 + 0.09} $ is:}

• A. $ 2 $
• B. $ 3 $
• C. $ 2.327 $
• D. $ 2.273 $

Hint:

When simplifying expressions involving powers and sums, look for patterns or formulas like the difference of cubes. Breaking down terms and simplifying step-by-step ensures accuracy.

Answer 1

The Correct Option is A

Step 1: Simplify the Numerator.
The numerator is: \[ (2.3)^3 - 0.027 \] Notice that \( 0.027 = (0.3)^3 \). So the numerator becomes: \[ (2.3)^3 - (0.3)^3 \] Use the identity: \[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \] Here, \( x = 2.3 \) and \( y = 0.3 \). So: \[ (2.3)^3 - (0.3)^3 = (2.3 - 0.3)((2.3)^2 + (2.3)(0.3) + (0.3)^2) \] Now simplify each term: \[ 2.3 - 0.3 = 2 \] \[ (2.3)^2 + (2.3)(0.3) + (0.3)^2 = 5.29 + 0.69 + 0.09 = 6 \] So the numerator is: \[ 2 \times 6 = 12 \] Step 2: Simplify the Denominator.
The denominator is: \[ (2.3)^2 + 0.69 + 0.09 = 5.29 + 0.78 = 6 \] Step 3: Compute the Fraction.
Now evaluate: \[ \frac{(2.3)^3 - 0.027}{(2.3)^2 + 0.69 + 0.09} = \frac{12}{6} = 2 \] Step 4: Analyze the Options.
Option (1): \( 2 \) — Correct
Option (2): \( 3 \) — Incorrect
Option (3): \( 2.327 \) — Incorrect
Option (4): \( 2.273 \) — Incorrect Step 5: Final Answer.
\[ (1) \quad \mathbf{2} \]