Evaluate the given expression.$$\left(\frac{-2}{5} ight)^{-2} imes \left(\frac{-4}{5} ight)^{4}$$

Question

Evaluate the given expression.$$\left(\frac{-2}{5}ight)^{-2} 	imes \left(\frac{-4}{5}ight)^{4}$$

Examoogle · Accepted Answer

The correct answer is Option C: $$\frac{64}{25}$$. Convert the negative exponent to positive by taking the reciprocal, simplify both parts: $(-5/2)^2 = 25/4$ and $(-4/5)^4 = 256/625$. Multiply them: $(25/4) 	imes (256/625) = (1/1) 	imes (64/25) = 64/25$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Convert the negative exponent to positive by taking the reciprocal, simplify both parts: $(-5/2)^2 = 25/4$ and $(-4/5)^4 = 256/625$. Multiply them: $(25/4) 	imes (256/625) = (1/1) 	imes (64/25) = 64/25$.

Formula Used: $$ (\frac{a}{b})^{-n} = (\frac{b}{a})^n $$ and $$ (\frac{a}{b})^m 	imes (\frac{c}{d})^n $$

Given: Expression to evaluate: (-2/5)^(-2) × (-4/5)^4

Steps: Convert negative exponent: Apply the rule $ (\frac{a}{b})^{-n} = (\frac{b}{a})^n $ to the first term $ (\frac{-2}{5})^{-2} $.. Evaluate the second term: Expand the second term $ (\frac{-4}{5})^4 $ knowing that an even power makes the negative base positive.. Multiply the results: Multiply the two simplified fractions $ \frac{25}{4} 	imes \frac{256}{625} $.

Answer: C is correct because the final evaluation of the expression yields 64/25.

Exam Tip: This type of simplification is fundamental for Algebra and Power/Exponent topics frequently appearing in railway recruitment exams.

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