$(0.1 \times 0.01 \times 0.001 \times 10^7)$ is equal to

### Concept & Formula Convert decimals into powers of 10 to easily apply the laws of exponents. When multiplying terms with the same base, add their exponents: $$a^m \times a^n = a^{m+n}$$ ### Step-by-Step Solution **Given:** $(0.1 \times 0.01 \times 0.001 \times 10^7)$ First, express each decimal as a power of 10: $0.1 = 10^{-1}$ $0.01 = 10^{-2}$ $0.001 = 10^{-3}$ Now, substitute these back into the original expression: $(10^{-1} \times 10^{-2} \times 10^{-3} \times 10^7)$ Apply the law of exponents by adding all the powers together: Exponent sum = $-1 + (-2) + (-3) + 7$ Exponent sum = $-6 + 7 = 1$ The final expression is $10^1$, which simplifies to $10$. ### Exam Strategy & Shortcut Count the total decimal places across all the fractional numbers. There is 1 place in $0.1$, 2 places in $0.01$, and 3 places in $0.001$. Total decimal places = $1 + 2 + 3 = 6$. This is equivalent to dividing by $10^6$. Multiplying by $10^7$ and dividing by $10^6$ leaves exactly $10^1$. ### Common Pitfall Trying to multiply the decimals directly to get $0.000001$ and then counting decimal places against the $10^7$ zeros is prone to miscounting. Converting to exponents immediately is much safer and faster. ### Final Answer Therefore, the correct answer is $10$.