Question

Express each of the following numbers in scientific (exponential) notation. a. 529 b. 240,000,000 c. 301,000,000,000,000,000 d. 78,444 e. 0.0003442 f. 0.000000000902 g. 0.043 h. 0.0821

Short answer

a. \(5.29 \times 10^2\) b. \(2.4 \times 10^8\) c. \(3.01 \times 10^{17}\) d. \(7.8444 \times 10^4\) e. \(3.442 \times 10^{-4}\) f. \(9.02 \times 10^{-10}\) g. \(4.3 \times 10^{-2}\) h. \(8.21 \times 10^{-2}\)

Step-by-step solution

Identify the first non-zero digit

Locate the first non-zero digit in the number: 5.

Create the decimal number

Place a decimal point after the first non-zero digit: 5.29

Determine the power of 10

Count the number of decimal places you moved (2), so the exponent is 2: \(5.29 \times 10^2\) b. 240,000,000

Identify the first non-zero digit

Locate the first non-zero digit in the number: 2.

Create the decimal number

Place a decimal point after the first non-zero digit: 2.4

Determine the power of 10

Count the number of decimal places you moved (8), so the exponent is 8: \(2.4 \times 10^8\) c. 301,000,000,000,000,000

Identify the first non-zero digit

Locate the first non-zero digit in the number: 3.

Create the decimal number

Place a decimal point after the first non-zero digit: 3.01

Determine the power of 10

Count the number of decimal places you moved (17), so the exponent is 17: \(3.01 \times 10^{17}\) d. 78,444

Identify the first non-zero digit

Locate the first non-zero digit in the number: 7.

Create the decimal number

Place a decimal point after the first non-zero digit: 7.8444

Determine the power of 10

Count the number of decimal places you moved (4), so the exponent is 4: \(7.8444 \times 10^4\) e. 0.0003442

Identify the first non-zero digit

Locate the first non-zero digit in the number: 3.

Create the decimal number

Place a decimal point after the first non-zero digit: 3.442

Determine the power of 10

Count the number of decimal places you moved (4), so the exponent is -4: \(3.442 \times 10^{-4}\) f. 0.000000000902

Identify the first non-zero digit

Locate the first non-zero digit in the number: 9.

Create the decimal number

Place a decimal point after the first non-zero digit: 9.02

Determine the power of 10

Count the number of decimal places you moved (10), so the exponent is -10: \(9.02 \times 10^{-10}\) g. 0.043

Identify the first non-zero digit

Locate the first non-zero digit in the number: 4.

Create the decimal number

Place a decimal point after the first non-zero digit: 4.3

Determine the power of 10

Count the number of decimal places you moved (2), so the exponent is -2: \(4.3 \times 10^{-2}\) h. 0.0821

Identify the first non-zero digit

Locate the first non-zero digit in the number: 8.

Create the decimal number

Place a decimal point after the first non-zero digit: 8.21

Determine the power of 10

Count the number of decimal places you moved (2), so the exponent is -2: \(8.21 \times 10^{-2}\)

Key concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of the same number. When you see a number like \(10^3\), this means \(10\) multiplied by itself three times: \(10 \times 10 \times 10\). Exponents have two parts: the base and the exponent itself. The base is the number that gets multiplied, and the exponent is the small number written above and to the right, indicating how many times to multiply the base by itself.

• For example, \(5^2\) represents \(5 \times 5\), which equals \(25\).
• \(2^4\) represents \(2 \times 2 \times 2 \times 2\), which equals \(16\).

Exponents are crucial in scientific notation because they allow us to represent very large and very small numbers in a more manageable form, allowing for easier computation and better clarity when dealing with various magnitudes of numbers.

Decimal Numbers Simplified

Decimal numbers are another way to represent fractions or numbers that are not whole. They use a decimal point to separate the whole number part from the fractional part. For example, the number \(3.25\) means \(3\) whole units and \(25\) hundredths of a unit.
Decimal points are vital when converting numbers to scientific notation as they help in isolating the significant digits of a number. Here's how it works:

• Move the decimal point so that only one non-zero digit remains on the left side. For instance, turning \(529\) into \(5.29\) puts it in the form needed for scientific notation.
• If you're dealing with a small number like \(0.0043\), similarly move the decimal point to just after the first non-zero digit, converting it to \(4.3\).

Decimal numbers thus help express the number in a compact form, making it easier to understand and work with them in scientific notation.

The Significance of Powers of Ten

Powers of ten are instrumental in scientific notation, as they help scale the numbers. The 'ten' is used because our number system is base ten, making it intuitive and practical for representation. Each power of ten shifts the decimal point: moving it to the right increases the number, while moving it to the left decreases it. For example:

• \(10^1\) equals \(10\), moving the decimal point one position to the right.
• \(10^{-1}\) equals \(0.1\), moving the decimal one position to the left.

In scientific notation, the power of ten reflects the number of spaces the decimal point has been moved from its original position. For large numbers, the exponent is positive, showing how many places the decimal has moved to the right. Conversely, for small numbers, the exponent is negative, indicating the decimal shift to the left. This method efficiently conveys the size and magnitude of a number, simplifying complex calculations.