Question

Express each number in scientific notation. \(\begin{array}{lllll}{\text { a. } 700} & {\text { c. } 4,500,000} & {\text { e. } 0.0054} & {\text { g. } 0.000000076} \\ {\text { b. } 38,000} & {\text { d. } 685,000,000,000} & {\text { f. } 0.00000687} & {\text { h. } 0.0000000008}\end{array}\)

Short answer

The numbers in scientific notation are as follows: a. \(700 = 7 \times 10^2\) b. \(38{,}000 = 3.8 \times 10^4\) c. \(4{,}500{,}000 = 4.5 \times 10^6\) d. \(685{,}000{,}000{,}000 = 6.85 \times 10^{11}\) e. \(0.0054 = 5.4 \times 10^{-3}\) f. \(0.00000687 = 6.87 \times 10^{-6}\) g. \(0.000000076 = 7.6 \times 10^{-8}\) h. \(0.0000000008 = 8 \times 10^{-10}\)

Step-by-step solution

a. Express 700 in scientific notation

To express 700 in scientific notation, we will write it as a product of a number between 1 and 10 and a power of ten. In this case, we have: \(700 = 7 \times 10^2\)

b. Express 38,000 in scientific notation

To express 38,000 in scientific notation, we will write it as a product of a number between 1 and 10 and a power of ten. In this case, we have: \(38{,}000 = 3.8 \times 10^4\)

c. Express 4,500,000 in scientific notation

To express 4,500,000 in scientific notation, we will write it as a product of a number between 1 and 10 and a power of ten. In this case, we have: \(4{,}500{,}000 = 4.5 \times 10^6\)

d. Express 685,000,000,000 in scientific notation

To express 685,000,000,000 in scientific notation, we will write it as a product of a number between 1 and 10 and a power of ten. In this case, we have: \(685{,}000{,}000{,}000 = 6.85 \times 10^{11}\)

e. Express 0.0054 in scientific notation

To express 0.0054 in scientific notation, we will write it as a product of a number between 1 and 10 and a power of ten. In this case, we have: \(0.0054 = 5.4 \times 10^{-3}\)

f. Express 0.00000687 in scientific notation

To express 0.00000687 in scientific notation, we will write it as a product of a number between 1 and 10 and a power of ten. In this case, we have: \(0.00000687 = 6.87 \times 10^{-6}\)

g. Express 0.000000076 in scientific notation

To express 0.000000076 in scientific notation, we will write it as a product of a number between 1 and 10 and a power of ten. In this case, we have: \(0.000000076 = 7.6 \times 10^{-8}\)

h. Express 0.0000000008 in scientific notation

To express 0.0000000008 in scientific notation, we will write it as a product of a number between 1 and 10 and a power of ten. In this case, we have: \(0.0000000008 = 8 \times 10^{-10}\)

Key concepts

Powers of Ten

In mathematics, the concept of "powers of ten" is essential, especially when dealing with large or small numbers. A power of ten means the number ten (10) raised to an exponent, signaling repeated multiplication. For instance, \(10^2 = 10 \times 10 = 100\).

The exponent indicates how many times the number ten is multiplied by itself. If the exponent is positive, like in \(10^3\), it represents the number 1000. Conversely, negative exponents like \(10^{-3}\) indicate a fraction, equal to \(1/10^3 = 0.001\). This is useful for representing very small numbers.

Working with powers of ten simplifies calculations and helps in expressing numbers in scientific notation. It reduces complex numeric expressions into simple base-ten relationships, allowing us to easily understand and manipulate them.

• Positive exponents: Larger numbers. E.g., \(10^6 = 1,000,000\)
• Negative exponents: Smaller numbers. E.g., \(10^{-6} = 0.000001\)

Standard Form

The term "standard form" refers to expressing a number as a decimal or integer without exponents. In scientific contexts, standard form often means using a combination of a number between 1 and 10 and a power of ten.

This way, numbers are neat and consistent, making them easier to compare and compute. For example, instead of writing 4500000, which can be cumbersome, one might express it in standard form as \(4.5 \times 10^6\).

The main advantage of standard form is its versatility, particularly in scientific equations, calculations, and data analysis.

• Simplifies large and small numbers
• Facilitates comparison and calculation
• Integral for scientific and engineering applications

Decimal Notation

Decimal notation is a system for representing numbers using the base ten; it consists of digits 0 through 9. It effectively handles both whole numbers and fractions, with a decimal point separating them.

Consider the number 38,000 in decimal notation: it's written directly as "38000". In contrast, smaller values like 0.0054 require careful placement of the decimal point.

Decimal notation is prevalent in everyday life for simple computations and record-keeping, making it accessible and universally accepted. It serves as a foundational block for both standard and scientific notations.

• Easy to understand and use
• Supports arithmetic and logical operations
• Forms basis for higher-level numeric expressions