Question

Write each of the following numbers in standard scientific notation. a. \(1 / 1033\) b. \(1 / 10^{5}\) c. \(1 / 10^{-7}\) d. \(1 / 0.0002\) e. \(1 / 3,093,000\) f. \(1 / 10^{-4}\) g. \(1 / 10^{9}\) h. \(1 / 0.000015\)

Short answer

a. \(9.6827 \times 10^{-4}\) b. \(1 \times 10^{-5}\) c. \(1 \times 10^7\) d. \(5 \times 10^3\) e. \(3.2366 \times 10^{-7}\) f. \(1 \times 10^{-4}\) g. \(1 \times 10^{-9}\) h. \(6.6667 \times 10^4\)

Step-by-step solution

Understand the goal of the task

The goal of this task is to convert the given numbers into standard scientific notation. To do this, we'll find a coefficient between 1 and 10 (including 1 but not 10), and an integer power of 10 so that when multiplied, the result is equal to the given number.

Convert each number to standard scientific notation

a. \(1 / 1033\): In this case, we divide 1 by 1033, which gives us 0.000968269746647. To write this in scientific notation, we can write it as \(9.68269746647 \times 10^{-4}\). b. \(1 / 10^5\): This number is already expressed as a power of 10, so we can write it directly as \(1 \times 10^{-5}\) in standard scientific notation. c. \(1 / 10^{-7}\): Notice that dividing by a negative power of 10 is the same as multiplying by its positive counterpart. Thus, this can be written as \(1 \times 10^7\) in standard scientific notation. d. \(1 / 0.0002\): In this case, we divide 1 by 0.0002, which gives us 5000. To write this in scientific notation, we can write it as \(5 \times 10^3\). e. \(1 / 3,093,000\): In this case, we divide 1 by 3,093,000, which gives us 0.0000003236574746. To write this in scientific notation, we can write it as \(3.236574746 \times 10^{-7}\). f. \(1 / 10^{4}\): This number is already expressed as a power of 10, so we can write it directly as \(1 \times 10^{-4}\) in standard scientific notation. g. \(1 / 10^9\): This number is already expressed as a power of 10, so we can write it directly as \(1 \times 10^{-9}\) in standard scientific notation. h. \(1 / 0.000015\): In this case, we divide 1 by 0.000015, which gives us 66666.66666666667. To write this in scientific notation, we can write it as \(6.666666666666667 \times 10^4\). The answers are: a. \(9.68269746647 \times 10^{-4}\) b. \(1 \times 10^{-5}\) c. \(1 \times 10^7\) d. \(5 \times 10^3\) e. \(3.236574746 \times 10^{-7}\) f. \(1 \times 10^{-4}\) g. \(1 \times 10^{-9}\) h. \(6.666666666666667 \times 10^4\)

Key concepts

Standard Scientific Notation

Understanding standard scientific notation is key to working with large or small numbers in a manageable way. This notation method simplifies numbers by condensing them into two parts: a coefficient and a power of ten. The coefficient must be a number greater than or equal to 1 and less than 10, and it's often a decimal. The second part is an exponent, which indicates how many times to multiply the coefficient by ten.

To convert a regular number to scientific notation, you need to move the decimal point to the right or left until only one non-zero digit remains on its left. The number of times you move the decimal point becomes the exponent, with rightward moves making the exponent negative, and leftward moves, positive. For example, converting 250,000 to scientific notation results in a coefficient of 2.5 and an exponent of 5, or written as: \(2.5 \times 10^5\).

Exponents

Exponents, also known as powers, are a shorthand way to express repeated multiplication of a number. When converting to scientific notation, the exponent tells us how many places to move the decimal point. For example, the exponent 3 in \(10^3\), means \(10 \times 10 \times 10\), which equals 1000. The exponent signifies the power of ten that, when multiplied by the coefficient, gives us the original number.

In the context of our exercise, when we have a negative exponent such as \(-4\) in \(9.68269746647 \times 10^{-4}\), it instructs us to move the decimal four places to the left. This transforms the scientific notation back to the standard form, 0.000968269746647, which is the original number we started with.

Powers of Ten

The concept of powers of ten is foundational for understanding scientific notation. A power of ten is simply 10 raised to an exponent, signifying the number of times ten multiplies itself. Positive exponents result in large values, whereas negative exponents produce small, fractional values. For example, \(10^3\) equals 1000, while \(10^{-3}\) equals 0.001.

When writing numbers in scientific notation, the powers of ten component greatly simplifies the representation of very large or very small numbers. In our exercise, converting numbers such as \(1 / 10^5\) directly to \(1 \times 10^{-5}\) showcases the simplicity and effectiveness of expressing numbers as powers of ten.

Notation Conversion

Notation conversion from standard form to scientific notation can seem daunting, but it follows a consistent process. Start by identifying the most significant digit in the standard form number and then determine the appropriate power of ten. When numbers are already expressed as a fraction of a power of ten, as seen in the textbook exercise, you can often convert it with minimal adjustments.

For example, taking \(1 / 0.000015\), we find the equivalent standard form number by dividing 1 by 0.000015, resulting in 66666.66666666667. In scientific notation, we express this result logically as \(6.666666666666667 \times 10^4\), showing a concise and normalized way of handling numbers regardless of their size, using the concept of powers of ten and a simple coefficient.