Find the value of the following expression : \(\frac{(0.22×0.22×0.22)+(0.022×0.…
2024
Find the value of the following expression :
\(\frac{(0.22×0.22×0.22)+(0.022×0.022×0.022)}{(0.66×0.66×0.66)+(0.066×0.066×0.066)}\)
Answer: C. \(\frac{1}{27}\) — ConceptWhen every term in a denominator is obtained by multiplying the corresponding numerator term by the same factor k before cubing, each denominator cube…
- A.
\(\frac{2}{27}\)
- B.
\(\frac{1}{25}\)
- C.
\(\frac{1}{27}\)
- D.
\(\frac{1}{23}\)
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Show answer & explanation
Correct answer: C
Concept
When every term in a denominator is obtained by multiplying the corresponding numerator term by the same factor k before cubing, each denominator cube is k3 times the corresponding numerator cube.
Therefore, the whole denominator is k3 times the numerator, and the ratio simplifies to 1/k3.
Application
Let N = (0.22)3 + (0.022)3.
Since 0.66 = 3 × 0.22 and 0.066 = 3 × 0.022, the denominator is D = (3 × 0.22)3 + (3 × 0.022)3.
Using (3a)3 = 27a3, D = 27(0.22)3 + 27(0.022)3 = 27N.
Hence N/D = N/(27N) = \(\frac{1}{27}\).
Cross-check
Directly, N = 0.010658648 and D = 0.287783496; multiplying N by 27 gives D exactly. Thus the value is \(\frac{1}{27}\).