Carry out the following multiplications and divisions,...

Carry out the following multiplications and divisions, expressing your answer in scientific notation to the correct number of significant figures. a. $\left(4.0 \times 10^4\right) \times\left(1.32 \times 10^8\right)$ b. $\left(2.23 \times 10^{-6}\right) \times\left(1.230 \times 10^{-2}\right)$ c. $\left(3.200 \times 10^7\right) \times\left(1.10 \times 10^{-2}\right) \times\left(2.3 \times 10^{-7}\right)$ d. $\frac{6.0 \times 10^{-5}}{3.0 \times 10^3}$ e. $\frac{5.132 \times 10^7}{1.12 \times 10^3}$ f. $\frac{\left(3.2 \times 10^2\right) \times\left(3.31 \times 10^6\right)}{\left(4.00 \times 10^{-3}\right) \times\left(2.0 \times 10^6\right)}$

Carry out the following multiplications and divisions, expressing your answer in scientific notation to the correct number of significant figures.
a. $\left(4.0 \times 10^4\right) \times\left(1.32 \times 10^8\right)$
b. $\left(2.23 \times 10^{-6}\right) \times\left(1.230 \times 10^{-2}\right)$
c. $\left(3.200 \times 10^7\right) \times\left(1.10 \times 10^{-2}\right) \times\left(2.3 \times 10^{-7}\right)$
d. $\frac{6.0 \times 10^{-5}}{3.0 \times 10^3}$
e. $\frac{5.132 \times 10^7}{1.12 \times 10^3}$
f. $\frac{\left(3.2 \times 10^2\right) \times\left(3.31 \times 10^6\right)}{\left(4.00 \times 10^{-3}\right) \times\left(2.0 \times 10^6\right)}$

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Key Concepts

Scientific Notation

Scientific notation expresses any number as a product of a coefficient (a number typically between 1 and 10) and a power of ten. This system is particularly useful for representing very large or very small numbers in a simplified, standard form.

Significant Figures

Significant figures indicate the precision of a measured or calculated quantity. When performing operations like multiplication or division, the final result must be rounded to match the number of significant figures of the least precise measurement.

Multiplication and Division with Scientific Notation

In scientific notation, multiplication involves multiplying the coefficients and adding the exponents, while division consists of dividing the coefficients and subtracting the exponents. This approach streamlines calculations with powers of ten.

Normalization and Rounding in Scientific Notation

After carrying out calculations, the coefficient must be adjusted so that it falls between 1 and 10, known as normalization. The result is then rounded according to the rules of significant figures to accurately reflect the precision of the original values.

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