Question

evaluate the expression:
$(10^{-1} cdot 5^{-2})^2$

Explanation:

Step1: Apply power of a product rule

$(ab)^n=a^n b^n$, so:
$$(10^{-1} \cdot 5^{-2})^2 = (10^{-1})^2 \cdot (5^{-2})^2$$

Step2: Apply power of a power rule

$(a^m)^n=a^{m \cdot n}$, so:
$$(10^{-1})^2 \cdot (5^{-2})^2 = 10^{-2} \cdot 5^{-4}$$

Step3: Convert negative exponents to fractions

$a^{-n}=\frac{1}{a^n}$, so:
$$10^{-2} \cdot 5^{-4} = \frac{1}{10^2} \cdot \frac{1}{5^4}$$

Step4: Calculate positive powers

$10^2=100$, $5^4=625$, so:
$$\frac{1}{100} \cdot \frac{1}{625} = \frac{1}{100 \times 625}$$

Step5: Multiply denominators

$$\frac{1}{100 \times 625} = \frac{1}{62500}$$

Answer:

$\frac{1}{62500}$ or $0.000016$