$(0.04)^{-1.5} = x$

### Concept & Formula This problem tests two key concepts: handling negative exponents and converting decimals to fractions. A negative exponent indicates the reciprocal of the base. $$ a^{-m} = \frac{1}{a^m} = \left(\frac{1}{a}\right)^m $$ ### Step-by-Step Solution * Given the expression: $(0.04)^{-1.5}$ * First, convert the decimal base into a fraction to make it easier to manipulate. * $0.04 = \frac{4}{100} = \frac{1}{25}$ * Substitute this back into the expression: $\left(\frac{1}{25}\right)^{-1.5}$ * Apply the negative exponent rule to flip the fraction and make the exponent positive: $(25)^{1.5}$ * Convert the decimal exponent to a fraction: $1.5 = \frac{3}{2}$. * The expression is now: $(25)^{\frac{3}{2}}$ * This means taking the square root of $25$ and then cubing the result. Since $25 = 5^2$: * $(5^2)^{\frac{3}{2}} = 5^{2 \times \frac{3}{2}} = 5^3$ * Calculate $5^3 = 5 \times 5 \times 5 = 125$. ### Exam Strategy & Shortcut Read $0.04$ as $\frac{1}{25}$. A negative power flips it to $25$. A power of $1.5$ means "the number times its square root". So, $25 \times \sqrt{25} = 25 \times 5 = 125$. This avoids calculating $5^3$ and is extremely fast to execute mentally. ### Common Pitfall A common mistake is mishandling the negative sign in the exponent. Some might mistakenly multiply the base by a negative value or fail to invert the fraction properly, resulting in a tiny decimal answer rather than the correct whole number. ### Final Answer **Therefore, the correct answer is 125.**