- NCL (Northern Coalfields Limited)-NO

NCL (Northern Coalfields Limited)-NO

Non Nursing Subjects

Hard

Appeared in: NCL (Northern Coalfields Limited)-NO

Explanation

The core task is to solve for 'm' in an equation involving exponents.
The key is to first standardize all terms to a common base. Using the rule (a/b)^x = (b/a)^-x, we convert terms with base (16/10) to base (10/16).
(16/10)^10 becomes (10/16)^-10, and (16/10)^16 becomes (10/16)^-16.
Using the rule a^x * a^y = a^(x+y), the exponents on the left side are added: (-10) + 7 + (-16) = -19.
The equation simplifies to (10/16)^-19 = (10/16)^(3m+6).
By equating the exponents, we get -19 = 3m + 6, which solves to 3m = -25, and therefore m = -25/3.

Option B: This value would be obtained if the sum of the exponents on the left side was incorrectly calculated as -10. For example, making a mistake in adding -19 and -7. The correct sum is -10 + 7 - 16 = -19.
Option C: This value might result from multiple calculation errors. For m = -31/3, the right-side exponent would be 3*(-31/3) + 6 = -31 + 6 = -25. This would require the left-side exponent to be -25, not -19.
Option D: This value might arise from an arithmetic error. For m = -29/3, the right-side exponent would be 3*(-29/3) + 6 = -29 + 6 = -23. This would require the left-side exponent to be -23, which is not the case.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Solving exponential equations using laws of exponents as background academic context rather than a clinical decision trigger.
This question assesses general aptitude and mathematical skills, which are part of many nursing entrance examinations.
While not directly related to clinical practice, strong analytical and problem-solving skills are essential for nurses in areas like medication dosage calculation, interpreting lab results, and managing patient data.
What if the middle term was also (16/10)^7? The equation would be (16/10)^10 × (16/10)^7 × (16/10)^16. All bases would be the same, so you would add the exponents: 10 + 7 + 16 = 33. The equation would be (16/10)^33 = (10/16)^(3m+6), which is (16/10)^33 = (16/10)^-(3m+6). Then 33 = -3m - 6, so 39 = -3m, and m = -13.

How to Approach the Question

First, analyze the equation and notice that it involves terms with exponents.
Identify that the bases of the terms, (16/10) and (10/16), are reciprocals of each other.
Choose a common base for all terms. It's often easiest to match the base on the side of the equation with the variable (in this case, 10/16).
Apply the exponent rule (a/b)^x = (b/a)^-x to convert all terms to the chosen common base.
Once all terms have the same base, use the rule a^m × a^n = a^(m+n) to simplify the side with multiple terms by adding their exponents.
After simplifying, you will have an equation of the form b^x = b^y. At this point, you can equate the exponents: x = y.

Concept Tested & Keywords

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