Question

Express each of the following numbers in standard scientific notation. a. 9,367,421 b. 7241 c. 0.0005519 d. 5.408 e. \(6.24 \times 10^{2}\) f. \(6319 \times 10^{-2}\) g. 0.000000007215 h. 0.721

Short answer

The short answer for the given numbers in standard scientific notation are: a. \(9.367421 \times 10^6\) b. \(7.241 \times 10^3\) c. \(5.519 \times 10^{-4}\) d. \(5.408 \times 10^0\) e. \(6.24 \times 10^2\) (already in standard scientific notation) f. \(6.319 \times 10^1\) g. \(7.215 \times 10^{-9}\) h. \(7.21 \times 10^{-1}\)

Step-by-step solution

Expressing 9,367,421 in standard scientific notation

To express 9,367,421 in standard scientific notation, we need to find a number between 1 and 10 and a power of 10 such that their product equals 9,367,421. Divide 9,367,421 by 10^6 (or count the number of digits to the right of the first digit) to get 9.367421, so the answer is \(9.367421 \times 10^6\).

Expressing 7241 in standard scientific notation

To express 7241 in standard scientific notation, we need to find a number between 1 and 10 and a power of 10 such that their product equals 7241. Divide 7241 by 10^3 (or count the number of digits to the right of the first digit) to get 7.241, so the answer is \(7.241 \times 10^3\).

Expressing 0.0005519 in standard scientific notation

To express 0.0005519 in standard scientific notation, we need to find a number between 1 and 10 and a negative power of 10 such that their product equals 0.0005519. Count the number of decimal places between the first nonzero digit (5) and the decimal point and multiply 0.0005519 by 10^4 to get 5.519. This gives us \(5.519 \times 10^{-4}\).

Expressing 5.408 in standard scientific notation

The number 5.408 is already in standard scientific notation, as it is between 1 and 10 and multiplied by a power of 10. If we want to express it more explicitly, we can write it as \(5.408 \times 10^0\), since any nonzero number raised to the power of 0 is 1.

Rearranging \(6.24 \times 10^2\)

This number is already in standard scientific notation, so no rearrangement is needed.

Rearranging \(6319 \times 10^{-2}\)

To express this number in standard scientific notation, divide 6319 by 10^3 (or count the number of digits to the right of the first digit) to get 6.319 and then add the exponent from the power of 10 to the existing exponent: -2 + 3 = 1. The answer is \(6.319 \times 10^1\).

Expressing 0.000000007215 in standard scientific notation

To express 0.000000007215 in standard scientific notation, we need to find a number between 1 and 10 and a negative power of 10 such that their product equals 0.000000007215. Count the number of decimal places between the first nonzero digit (7) and the decimal point and multiply 0.000000007215 by 10^9 to get 7.215. This gives us \(7.215 \times 10^{-9}\).

Expressing 0.721 in standard scientific notation

To express 0.721 in standard scientific notation, we need to find a number between 1 and 10 and a negative power of 10 such that their product equals 0.721. Since 0.721 is already between 1 and 10, we just need to express the given number with the appropriate power of 10. We have one decimal place to the right of the first digit, so the answer is \(7.21 \times 10^{-1}\).

Key concepts

Standard Scientific Notation

Scientific notation is an efficient way to express very large or very small numbers. It is often used in science and mathematics to make calculations with such numbers more manageable.

In standard scientific notation, a number is written as the product of two factors. The first is a decimal greater than or equal to 1 and less than 10, and the second is an integer power of 10. For example, the number 7241 is represented as 7.241 multiplied by 10 to the third power, or \(7.241 \times 10^3\).

To convert a number to this format, you start with the original number and move the decimal point until you have a number between 1 and 10. The number of places you move the decimal point becomes the exponent on the 10. If you move the decimal point to the left, the exponent is positive; if you move it to the right, the exponent is negative.

Utilizing proper scientific notation ensures brevity and precision, especially when handling calculations that involve either extremely large or diminutive numbers.

Exponents in Scientific Notation

Understanding exponents is crucial when expressing numbers in scientific notation. An exponent in scientific notation indicates the power of 10 required to transform a number back to its original value.

For instance, when we write \(6.319 \times 10^1\), the exponent 1 tells us that the decimal point in 6.319 needs to be moved one place to the right to get back the original number, which is 63.19. If the exponent were negative, as in \(5.519 \times 10^{-4}\), we would move the decimal point four places to the left, denoting a number much smaller than one.

Using exponents allows us to express large scales of numbers compactly and perform arithmetic operations without directly dealing with long strings of zeros.

Expressing Numbers in Scientific Notation

The process of expressing numbers in scientific notation may seem daunting at first, but it follows a simple set of steps that can be applied to any number, regardless of its size.

For a large number like 9,367,421, count the digits to the right of the first significant digit and move the decimal after the first digit to get 9.367421, and then denote the shift with an exponent on 10: \(9.367421 \times 10^6\).

For small numbers like 0.000000007215, count the decimal places before the first significant digit to determine the exponent. In this case, it means moving the decimal 9 places to the right, resulting in \(7.215 \times 10^{-9}\).

Mastering the ability to express numbers in scientific notation is not only a useful mathematical skill but also a critical tool for scientific measurement, data analysis, engineering, and beyond, as it allows for a clearer understanding and more straightforward calculations of very large and very small quantities.