Question

Express each of the following as an "ordinary" decimal number. a. \(3.011 \times 10^{23}\) b. \(5.091 \times 10^{9}\) c. \(7.2 \times 10^{2}\) d. \(1.234 \times 10^{5}\) e. \(4.32002 \times 10^{-4}\) f. \(3.001 \times 10^{-2}\) g. \(2.9901 \times 10^{-7}\) h. \(4.2 \times 10^{-1}\)

Short answer

a. \(301100000000000000000000\) b. \(5091000000\) c. \(720\) d. \(123400\) e. \(0.000432002\) f. \(0.03001\) g. \(0.00000029901\) h. \(0.42\)

Step-by-step solution

Identify the exponent

The exponent is 23 in this case.

Move the decimal point

Since the exponent is positive, we will move the decimal point 23 places to the right. So the answer is: \(301100000000000000000000\). b. Express \(5.091 \times 10^{9}\) as an ordinary decimal number.

Identify the exponent

The exponent is 9 in this case.

Move the decimal point

Since the exponent is positive, we will move the decimal point 9 places to the right. So the answer is: \(5091000000\). c. Express \(7.2 \times 10^{2}\) as an ordinary decimal number.

Identify the exponent

The exponent is 2 in this case.

Move the decimal point

Since the exponent is positive, we will move the decimal point 2 places to the right. So the answer is: \(720\). d. Express \(1.234 \times 10^{5}\) as an ordinary decimal number.

Identify the exponent

The exponent is 5 in this case.

Move the decimal point

Since the exponent is positive, we will move the decimal point 5 places to the right. So the answer is: \(123400\). e. Express \(4.32002 \times 10^{-4}\) as an ordinary decimal number.

Identify the exponent

The exponent is -4 in this case.

Move the decimal point

Since the exponent is negative, we will move the decimal point 4 places to the left. So the answer is: \(0.000432002\). f. Express \(3.001 \times 10^{-2}\) as an ordinary decimal number.

Identify the exponent

The exponent is -2 in this case.

Move the decimal point

Since the exponent is negative, we will move the decimal point 2 places to the left. So the answer is: \(0.03001\). g. Express \(2.9901 \times 10^{-7}\) as an ordinary decimal number.

Identify the exponent

The exponent is -7 in this case.

Move the decimal point

Since the exponent is negative, we will move the decimal point 7 places to the left. So the answer is: \(0.00000029901\). h. Express \(4.2 \times 10^{-1}\) as an ordinary decimal number.

Identify the exponent

The exponent is -1 in this case.

Move the decimal point

Since the exponent is negative, we will move the decimal point 1 place to the left. So the answer is: \(0.42\).

Key concepts

Decimal Numbers

Decimal numbers, a fundamental concept in mathematics, are a way of expressing numbers that include a decimal point to represent a fraction of a base 10 number. They are composed of two parts: the whole number part and the fractional part, separated by a decimal point (.). For instance, in the number 45.678, '45' is the whole number part and '678' is the fractional part.

Decimal numbers can be expanded to represent very large or very small values. When understanding how to express decimal numbers in 'ordinary' form from scientific notation, it's essential to grasp the role of the decimal point. Movement of the decimal point to the right increases the value of the number, indicating multiplication by a power of 10 for every place moved. Conversely, moving the decimal to the left decreases the value, reflecting division by a power of 10 per each place moved.

Exponents

Exponents, also known as powers, are a shorthand notation in mathematics for expressing repeated multiplication of a number by itself. An exponent is written as a small superscript number to the top right of a base number, like in \(2^3\). This tells us to multiply the base number, 2, by itself the number of times indicated by the exponent, 3, resulting in \(2 \times 2 \times 2 = 8\).

In the context of decimal numbers, when we have a positive exponent, the base number is enlarged by being multiplied by 10 for each unit in the exponent. For example, \(3.011 \times 10^{23}\) signifies that 3.011 should be multiplied by 10, 23 times over. In the exercises, when the exponent is negative, such as in \(4.32002 \times 10^{-4}\), it indicates that the decimal point in the base number should be moved to the left, decreasing its value, as we effectively divide by 10 for each negative exponent value.

Scientific Notation Conversion

Converting scientific notation to ordinary decimal form is a skill that allows us to represent extremely large or small numbers in a more readable format. Scientific notation is written as a product of two numbers: a coefficient (typically between 1 and 10) followed by \(\times 10\) raised to an exponent. The conversion involves moving the decimal point in the coefficient to the right or left, depending on whether the exponent is positive or negative. The number of places the decimal point is moved corresponds to the absolute value of the exponent.

For example, to convert \(5.091 \times 10^9\) to an ordinary decimal number, we move the decimal point 9 places to the right because the exponent is positive, giving us 5091000000. If we were to convert \(3.001 \times 10^{-2}\), we would move the decimal point 2 places to the left, since the exponent is negative, resulting in 0.03001. Remember, if the decimal point is moved to the right, we may need to add zeros to fill in places, whereas moving to the left may involve placing the decimal point before existing digits.