Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$\color{blue}{8 \times 10^5}$
RECALL: The reciprocal of $a$ is $\dfrac{1}{a}$. Thus, the reciprocal of $1.25 \times 10^{-6}$ is $\dfrac{1}{1.25 \times 10^{-6}}$. Use the negative exponent rule ($a^{-m}=\dfrac{1}{a^m}, m\ne0$ to obtain: $\dfrac{1}{1.25 \times 10^{-6}} \\=\dfrac{1}{1.25 \times \frac{1}{10^6}} \\=\dfrac{1}{1.25} \cdot \dfrac{10^6}{1} \\=\dfrac{1 \times 10^6}{1.25} \\=\dfrac{10^6}{1.25}$ Since $1.25 = \dfrac{5}{4}$, the expression above is equivalent to: $=\dfrac{10^6}{\frac{5}{4}} \\=10^6 \times \dfrac{4}{5} \\=\dfrac{4}{5} \times 10^6$ With $\dfrac{4}{5}=0.8$, the expression above is equivalent to: $=0.8 \times 10^6$ Since the constant (non-power of 10) part of a scientific notation must be greater than or equal to 1 but less than 10, write $0.8$ as $8 \times 10^{-1}$ to obtain: $=8 \times 10^{-1} \times 10^6$ Using the rule $a^m \cdot a^n = a^{m+n}$, the expression above simplifies to: $=8 \times 10^{-1+6} \\=\color{blue}{8 \times 10^5}$