$$ (0.2 \times 0.2 + 0.01) (0.1 \times 0.1 + 0.02)^{-1} $$ is equal to

### Concept & Formula This problem tests the basic order of operations (BODMAS/PEMDAS) combined with the negative exponent rule. Remember that any expression raised to the power of $-1$ is equivalent to the reciprocal of that expression: $$ x^{-1} = \frac{1}{x} $$ ### Step-by-Step Solution **Given:** $$ (0.2 \times 0.2 + 0.01) (0.1 \times 0.1 + 0.02)^{-1} $$ **Calculation:** * First, evaluate the expression inside the first bracket: 1. Multiply: $0.2 \times 0.2 = 0.04$ 2. Add: $0.04 + 0.01 = 0.05$ So, the first bracket evaluates to $0.05$. * Next, evaluate the expression inside the second bracket: 1. Multiply: $0.1 \times 0.1 = 0.01$ 2. Add: $0.01 + 0.02 = 0.03$ So, the second bracket evaluates to $0.03$. * Substitute these resolved values back into the original equation: $$ (0.05) \times (0.03)^{-1} $$ * Apply the negative exponent rule: $$ (0.03)^{-1} = \frac{1}{0.03} $$ * Multiply the terms together, which acts as division: $$ \frac{0.05}{0.03} $$ * To simplify, multiply both the numerator and the denominator by $100$ to clear the decimals: $$ \frac{5}{3} $$ ### Exam Strategy & Shortcut **Fraction Conversion:** Instead of dealing with decimals, convert to fractions immediately if you are comfortable with them. $0.2 = 2/10$. So $(2/10 \times 2/10) + 1/100 = 4/100 + 1/100 = 5/100$. $0.1 = 1/10$. So $(1/10 \times 1/10) + 2/100 = 1/100 + 2/100 = 3/100$. The equation becomes: $(5/100) \times (3/100)^{-1}$ $(5/100) \times (100/3) = 5/3$. This bypasses decimal math entirely and prevents placement errors. ### Common Pitfall Incorrectly calculating the squares of decimals. A very common mistake is calculating $0.2 \times 0.2$ as $0.4$ instead of $0.04$. Always remember that multiplying two numbers with one decimal place each results in an answer with two decimal places. ### Final Answer **Therefore, the correct answer is 5/3.**