If 25^(-2t) = 1/(sqrt(125)), what is the value of t?-{3/8}3/83/48/3

1. TRANSLATE the problem information

• Given equation: \(25^{-2\mathrm{t}} = \frac{1}{\sqrt{125}}\)
• Need to find: value of t

2. INFER the solution strategy

• Key insight: When bases are different, express everything using the same base
• Since both 25 and 125 are powers of 5, we'll use base 5
• This will allow us to equate exponents directly

3. TRANSLATE numbers to the same base

• Convert \(25 = 5^2\)
• Convert \(125 = 5^3\), so \(\sqrt{125} = (5^3)^{1/2} = 5^{3/2}\)
• Rewrite equation: \((5^2)^{-2\mathrm{t}} = \frac{1}{5^{3/2}}\)

4. SIMPLIFY the exponential expressions

• Left side: \((5^2)^{-2\mathrm{t}} = 5^{-4\mathrm{t}}\)
• Right side: \(\frac{1}{5^{3/2}} = 5^{-3/2}\)
• New equation: \(5^{-4\mathrm{t}} = 5^{-3/2}\)

5. INFER the final step

• Since bases are identical, exponents must be equal
• Therefore: \(-4\mathrm{t} = -\frac{3}{2}\)

6. SIMPLIFY to solve for t

• Divide both sides by -4: \(\mathrm{t} = \frac{-3/2}{-4} = \frac{3}{8}\)

Answer: B) 3/8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the strategy of using a common base, instead trying to work directly with 25 and 125 as separate entities. They might attempt to convert everything to decimals or get stuck trying to manipulate the equation without a clear strategy.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor TRANSLATE execution: Students incorrectly convert \(\sqrt{125}\) to exponential form. They might write \(\sqrt{125} = 125^{1/2}\) but fail to simplify this as \(5^{3/2}\), or make errors in the negative exponent conversion (writing \(\frac{1}{5^{3/2}}\) as \(5^{3/2}\) instead of \(5^{-3/2}\)).

This creates a wrong equation, leading them to select Choice A (-3/8) or other incorrect values.

The Bottom Line:

This problem tests whether students can strategically choose a common base and systematically apply exponential laws. Success requires both strategic insight and careful algebraic manipulation.