Question

Given below are two statements:
Statement I: \((243)^{0.16} ×(9)^{0.1}=0.3\)
Statement II: If , \(3.105 ×10^P=0.00239+0.000715,\) then \(P=-3.\)
In the light of the above statements choose the most appropriate answer from the options given below:

• A. Both Statement I and Statement II are correct
• B. Both Statement I and Statement II are incorrect
• C. Statement I is correct but Statement II is incorrect
• D. Statement I is incorrect but Statement II is correct

Answer 1

The Correct Option is D

Let's evaluate each of the statements individually to determine which are correct. 

1. Statement I: \( (243)^{0.16} \times (9)^{0.1} = 0.3 \)
   • Firstly, express 243 and 9 in terms of powers of 3:
   • \(243 = 3^5\) and \(9 = 3^2\).
   • Substituting in the equation:
   • \((3^5)^{0.16} \times (3^2)^{0.1}\)
   • Simplifying the exponents using the property \((a^m)^n = a^{mn}\):
   • \(3^{0.8} \times 3^{0.2} = 3^{(0.8+0.2)} = 3^1\).
   • This means \(3^1 = 3\), not 0.3. Therefore, Statement I is incorrect.
2. Statement II: If \(3.105 \times 10^P = 0.00239 + 0.000715\), then \(P = -3\).
   • Calculate \(0.00239 + 0.000715\):
   • Sum: \(0.00239 + 0.000715 = 0.003105\).
   • Set \(3.105 \times 10^P = 0.003105\).
   • Divide both sides by 3.105:
   • \(10^P = \frac{0.003105}{3.105} = 0.001 = 10^{-3}\).
   • This implies \(P = -3\). Therefore, Statement II is correct.

Based on the above evaluations:

• Statement I is incorrect.
• Statement II is correct.

The most appropriate answer is, therefore: Statement I is incorrect but Statement II is correct.