Question

Express each of the following numbers in standard scientific notation. a. 0.005219 b. 5219 c. 6,199,291 d. 0.1973 e. 93,000,000 f. \(72.41 \times 10^{-2}\) g. \(0.007241 \times 10^{-5}\) h. 1.00

Short answer

a. 5.219 × 10\(^{-3}\) b. 5.219 × 10\(^{3}\) c. 6.199291 × 10\(^{6}\) d. 1.973 × 10\(^{-1}\) e. 9.3 × 10\(^{7}\) f. 7.241 × 10\(^{-1}\) g. 7.241 × 10\(^{-6}\) h. 1 × 10\(^{0}\)

Step-by-step solution

Convert to scientific notation with positive exponent

First, let's convert the numbers with positive exponents. For this, we will find the number of places we need to move the decimal point so that the result is a decimal number between 1 and 10. a. 0.005219 -> 5.219 × 10^(-3) b. 5219 -> 5.219 × 10^(3) c. 6,199,291 -> 6.199291 × 10^(6) e. 93,000,000 -> 9.3 × 10^(7)

Convert numbers given in the product of two numbers

For numbers given in the product of two numbers, you have to multiply the decimal number and the exponent, adding the exponents of the powers of 10 together. f. \(72.41 \times 10^{-2}\) -> \(7.241 \times 10^(-2+1)\) -> 7.241 × 10^(-1) g. \(0.007241 \times 10^{-5}\) -> \(7.241 \times 10^{(-1-5)}\) -> 7.241 × 10^(-6)

Convert numbers close to the standard format

For numbers close to the standard format, we just have to find the right exponent for the power of 10. d. 0.1973 -> 1.973 × 10^(-1) h. 1.00 -> 1 × 10^(0)

Final Answers

Here is the list of the numbers in standard scientific notation: a. 5.219 × 10^(-3) b. 5.219 × 10^(3) c. 6.199291 × 10^(6) d. 1.973 × 10^(-1) e. 9.3 × 10^(7) f. 7.241 × 10^(-1) g. 7.241 × 10^(-6) h. 1 × 10^(0)

Key concepts

Expressing Numbers in Scientific Notation

When numbers are very large or very small, expressing them in scientific notation makes them easier to work with. This format is composed of two main parts: a coefficient and an exponent. The coefficient is a number between 1 and 10, and the exponent is a power of ten that scales the number up or down.
For example, to express 0.005219 in scientific notation, locate the decimal point to give a coefficient between 1 and 10, which gives us 5.219. Since we moved the decimal three places to the right, we use a negative exponent, resulting in the scientific notation of 5.219 × 10^(-3). To convert 5219 into scientific notation, we move the decimal three places to the left, so the exponent is positive: 5.219 × 10^(3).

• Move the decimal point in the number until you have a coefficient between 1 and 10.
• Count the number of places you moved the decimal: this is your exponent.
• If you moved the decimal to the left, the exponent is positive; to the right, it's negative.
• Combine the coefficient and the exponent to express in scientific notation.

Scientific Notation Conversions

Converting between scientific notation and standard form is a key skill. If you're given a number like 72.41 × 10^(-2), it's already closely resembling scientific notation but isn't quite there. To convert, we adjust the coefficient: move the decimal one place to the left which subtracts one from the exponent. Our conversion would be 7.241 × 10^(-1).
Similarly, for 0.007241 × 10^(-5), move the decimal three places to the right to make the coefficient 7.241, and adjust the exponent from -5 to -1-5 or -6, resulting in 7.241 × 10^(-6).

Steps for Converting Scientific Notation

• Adjust the coefficient to be between 1 and 10 by moving the decimal point.
• Add or subtract the number of places you moved the decimal from the existing exponent.
• The final expression will have your new coefficient and adjusted exponent.

Exponents and Powers of Ten

The exponent in scientific notation is incredibly important as it indicates the scale of the number. A positive exponent, such as 10^(3) for 5219, means the number is large, telling us the decimal moves to the right. On the other hand, a negative exponent, like 10^(-3) for 0.005219, denotes a small number, where the decimal moves to the left.
Think of the exponent as the number of jumps the decimal makes. For every positive step, the number gets ten times bigger. Conversely, for each negative step, it gets ten times smaller.

Power of Ten

• Positive exponent: Moves the decimal to the right (larger number)
• Negative exponent: Moves the decimal to the left (smaller number)
• Each exponent step changes the size of the number by a factor of ten

Remember, an exponent of zero, as in 1 × 10^(0), means the number remains the same, because any number to the power of zero is 1.

Decimal Number Representation

In a decimal system, every digit has a value based on its position relative to the decimal point. The place to the left of the decimal indicates 'ones', the next place 'tens', then 'hundreds', and so forth. To the right, it's the opposite: 'tenths', 'hundredths', 'thousandths', etc.
For large numbers, this system can become cumbersome. That's where scientific notation shines, as it simplifies these long strings of digits into a manageable format. A number such as 6,199,291 is cumbersome, but its scientific notation is a streamlined 6.199291 × 10^(6), telling us exactly how large it is through the power of ten.

• Each position in a decimal number has a specific value.
• Decimal numbers can be long and complex for very large or small values.
• Scientific notation simplifies this complexity into a format that clearly indicates a number's size.