Question

By how many places must the decimal point be moved, and in which direction, to convert each of the following to "ordinary" decimal numbers? a. \(4.311 \times 10^{6}\) b. \(7.895 \times 10^{-5}\) c. \(8.712 \times 10^{1}\) d. \(4.995 \times 10^{0}\) e. \(2.331 \times 10^{18}\) f. \(1.997 \times 10^{-16}\)

Short answer

a. Move the decimal point 6 places to the right: \(4,311,000\) b. Move the decimal point 5 places to the left: \(0.00007895\) c. Move the decimal point 1 place to the right: \(87.12\) d. The decimal point doesn't move: \(4.995\) e. Move the decimal point 18 places to the right: \(2,331,000,000,000,000,000\) f. Move the decimal point 16 places to the left: \(0.0000000000000001997\)

Step-by-step solution

Identify the direction of movement

Since the exponent is positive, the decimal point should be moved to the right.

Move the decimal point

Move the decimal point 6 places to the right to convert to an ordinary decimal number: \(4.311 \times 10^6 = 4,311,000\) #b. \(7.895 \times 10^{-5}\)

Identify the direction of movement

Since the exponent is negative, the decimal point should be moved to the left.

Move the decimal point

Move the decimal point 5 places to the left to convert to an ordinary decimal number: \(7.895 \times 10^{-5} = 0.00007895\) #c. \(8.712 \times 10^{1}\)

Identify the direction of movement

Since the exponent is positive, the decimal point should be moved to the right.

Move the decimal point

Move the decimal point 1 place to the right to convert to an ordinary decimal number: \(8.712 \times 10^1 = 87.12\) #d. \(4.995 \times 10^{0}\)

Identify the direction of movement

Since the exponent is 0, the decimal point should not be moved.

Convert to an ordinary decimal number

Any number raised to the power of 0 is 1, so \(4.995 \times 10^0 = 4.995\) #e. \(2.331 \times 10^{18}\)

Identify the direction of movement

Since the exponent is positive, the decimal point should be moved to the right.

Move the decimal point

Move the decimal point 18 places to the right to convert to an ordinary decimal number: \(2.331 \times 10^{18} = 2,331,000,000,000,000,000\) #f. \(1.997 \times 10^{-16}\)

Identify the direction of movement

Since the exponent is negative, the decimal point should be moved to the left.

Move the decimal point

Move the decimal point 16 places to the left to convert to an ordinary decimal number: \(1.997 \times 10^{-16} = 0.0000000000000001997\)

Key concepts

Decimal Conversion

Decimal conversion is the process of changing numbers between scientific notation and standard decimal form. This is useful for expressing very large or very small numbers in a more manageable way. Scientific notation uses a base number, usually between 1 and 10, multiplied by a power of 10. For example, in the number \(4.311 \times 10^6\), 4.311 is the base number. To convert it to a regular decimal, you move the decimal point to the right or left, depending on the exponent. Scientific notation helps in simplifying comparisons and calculations with large or tiny numbers by focusing on the significant digits.

Positive Exponents

Positive exponents in scientific notation indicate how many places the decimal point has to be moved to the right. This transforms the base number into a larger number. For example, in the expression \(8.712 \times 10^1\), the exponent is 1, meaning you move the decimal one place to the right, resulting in 87.12. When the exponent is greater than 1, like in \(4.311 \times 10^6\), the decimal moves six places to the right, turning it into 4,311,000.

• Positive exponents multiply the base number, making it larger.
• They indicate the number of zeros to add at the end of a number (after adjusting decimal placement).

Negative Exponents

Negative exponents in scientific notation show how many places to move the decimal point to the left. This makes the base number smaller, often resulting in a number less than one. For instance, with \(7.895 \times 10^{-5}\), you move the decimal five places to the left, yielding 0.00007895. Similarly, \(1.997 \times 10^{-16}\) becomes 0.0000000000000001997 after moving the decimal 16 places left.

• Negative exponents divide the base number by powers of ten.
• They result in decimals that are very small numbers.
• Focus on the direction left for negative exponents.

Exponent Rules

Exponent rules are the guidelines that dictate how to work with powers of 10 in calculations. Understanding these can simplify arithmetic operations and conversions. Here are a few key ones:

• Any number raised to the power of 0 equals 1. This means \(4.995 \times 10^0\) remains 4.995, as the factor of 1 does not change the original value.
• When you multiply two powers of ten, you add their exponents: \(10^a \times 10^b = 10^{a+b}\).
• When dividing, you subtract the exponents: \(10^a / 10^b = 10^{a-b}\).

Understanding these rules helps simplify when converting and dealing with scientific notations and ensures accuracy in the processes.