By Fatskills Exam Guides Team — the exam nerds behind 28,500+ quizzes and 2.1M practice questions across 500+ global exams.
Scientific notation is a way of expressing very large or very small numbers in a compact and manageable form using a base of 10. It consists of a coefficient between 1 and 10 multiplied by a power of 10.
Scientific notation is essential in various fields, including physics, chemistry, engineering, and economics, where large or small numbers are common. For example, in physics, the speed of light is approximately 299,792,458 meters per second, which can be expressed in scientific notation as $2.99792458 \times 10^8$ meters per second. This notation allows for easier calculations and comparisons.
To write a number in scientific notation, move the decimal point to the left or right until you have a coefficient between 1 and 10. Then, multiply the coefficient by 10 raised to the power of the number of places you moved the decimal point.
$$ \begin{aligned} 456,789 &\approx 4.56789 \times 10^5 \0.000456 &\approx 4.56 \times 10^{-4} \end{aligned} $$
When adding or subtracting numbers in scientific notation, make sure they have the same exponent. If they don't, adjust the coefficient of the number with the smaller exponent to match the exponent of the other number.
$$ \begin{aligned} (3.4 \times 10^2) + (2.1 \times 10^2) &= (3.4 + 2.1) \times 10^2 \ &= 5.5 \times 10^2 \ (4.2 \times 10^3) - (2.1 \times 10^3) &= (4.2 - 2.1) \times 10^3 \ &= 2.1 \times 10^3 \end{aligned} $$
When multiplying or dividing numbers in scientific notation, multiply or divide the coefficients and add or subtract the exponents.
$$ \begin{aligned} (3.4 \times 10^2) \times (2.1 \times 10^3) &= (3.4 \times 2.1) \times 10^{2+3} \ &= 7.14 \times 10^5 \ (4.2 \times 10^3) \div (2.1 \times 10^2) &= \frac{4.2}{2.1} \times 10^{3-2} \ &= 2 \times 10^1 \end{aligned} $$
Express the number 456,789 in scientific notation.
Move the decimal point to the left until you have a coefficient between 1 and 10. In this case, you need to move the decimal point 5 places to the left.
$$ \begin{aligned} 456,789 &\approx 4.56789 \times 10^5 \end{aligned} $$
Add the numbers $(3.4 \times 10^2)$ and $(2.1 \times 10^2)$.
Since both numbers have the same exponent, you can add the coefficients directly.
$$ \begin{aligned} (3.4 \times 10^2) + (2.1 \times 10^2) &= (3.4 + 2.1) \times 10^2 \ &= 5.5 \times 10^2 \end{aligned} $$
Multiply the numbers $(4.2 \times 10^3)$ and $(2.1 \times 10^2)$.
Multiply the coefficients and add the exponents.
$$ \begin{aligned} (4.2 \times 10^3) \times (2.1 \times 10^2) &= (4.2 \times 2.1) \times 10^{3+2} \ &= 8.82 \times 10^5 \end{aligned} $$
What is the result of adding the numbers $(3.4 \times 10^2)$ and $(2.1 \times 10^2)$?
A) $5.5 \times 10^3$ B) $5.5 \times 10^2$ C) $3.4 \times 10^3$ D) $2.1 \times 10^3$
B) $5.5 \times 10^2$
A) $5.5 \times 10^3$ is a common mistake when adding numbers in scientific notation. Students may forget to add the exponents. C) $3.4 \times 10^3$ is a common mistake when adding numbers in scientific notation. Students may forget to add the coefficients. D) $2.1 \times 10^3$ is a common mistake when adding numbers in scientific notation. Students may forget to add the coefficients.
What is the result of multiplying the numbers $(4.2 \times 10^3)$ and $(2.1 \times 10^2)$?
A) $8.82 \times 10^5$ B) $8.82 \times 10^3$ C) $4.2 \times 10^5$ D) $2.1 \times 10^5$
A) $8.82 \times 10^5$
B) $8.82 \times 10^3$ is a common mistake when multiplying numbers in scientific notation. Students may forget to add the exponents. C) $4.2 \times 10^5$ is a common mistake when multiplying numbers in scientific notation. Students may forget to multiply the coefficients. D) $2.1 \times 10^5$ is a common mistake when multiplying numbers in scientific notation. Students may forget to multiply the coefficients.
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