Question:
If \((\sqrt{5})^3 \times 25^2 = 5^n \times 5\sqrt{5}\), then what is the value of n?
यदि \((\sqrt{5})^3 \times 25^2 = 5^n \times 5\sqrt{5}\) है, तो n का मान क्या है?
Options:
- 4
- 5
- 6
- 7
- 4
- 5
- 6
- 7
Correct Answer: a)
Explanation:
Expressing everything in powers of 5:
LHS: \((5^{1/2})^3 \times (5^2)^2 = 5^{3/2} \times 5^4 = 5^{11/2} = 5^{5.5}\).
RHS: \(5^n \times 5^1 \times 5^{1/2} = 5^{n + 1.5}\).
Equating exponents:
\(n + 1.5 = 5.5 \implies n = 4\).
5 की घातों में व्यक्त करने पर:
LHS = \(5^{1.5} \times 5^4 = 5^{5.5}\)।
RHS = \(5^{n + 1.5}\)।
घातों की तुलना करने पर: \(n + 1.5 = 5.5 \implies n = 4\)।