Question

Express each of the following as an "ordinary" decimal number. a. \(2.98 \times 10^{-5}\) b. \(4.358 \times 10^{9}\) c. \(1.9928 \times 10^{-6}\) d. \(6.02 \times 10^{23}\) e. \(1.01 \times 10^{-1}\) f. \(7.87 \times 10^{-3}\) g. \(9.87 \times 10^{7}\) h. \(3.7899 \times 10^{2}\) i. \(1.093 \times 10^{-1}\) j. \(2.9004 \times 10^{\circ}\) k. \(3.9 \times 10^{-4}\) 1\. \(1.904 \times 10^{-8}\)

Short answer

a. \(0.0000298\) b. \(4,358,000,000\) c. \(0.0000019928\) d. \(602,000,000,000,000,000,000,000\) e. \(0.101\) f. \(0.00787\) g. \(98,700,000\) h. \(378.99\) i. \(0.1093\) j. Error in the expression, please provide the correct exponent. k. \(0.00039\) 1. \(0.00000001904\)

Step-by-step solution

(a) Convert \(2.98 \times 10^{-5}\) to decimal form

As the exponent is -5, we will move the decimal point 5 places to the left: \(2.98 \times 10^{-5} = 0.0000298\)

(b) Convert \(4.358 \times 10^{9}\) to decimal form

As the exponent is 9, we will move the decimal point 9 places to the right: \(4.358 \times 10^9 = 4,358,000,000\)

(c) Convert \(1.9928 \times 10^{-6}\) to decimal form

As the exponent is -6, we will move the decimal point 6 places to the left: \(1.9928 \times 10^{-6} = 0.0000019928\)

(d) Convert \(6.02 \times 10^{23}\) to decimal form

As the exponent is 23, we will move the decimal point 23 places to the right: \(6.02 \times 10^{23} = 602,000,000,000,000,000,000,000\)

(e) Convert \(1.01 \times 10^{-1}\) to decimal form

As the exponent is -1, we will move the decimal point 1 place to the left: \(1.01 \times 10^{-1} = 0.101\)

(f) Convert \(7.87 \times 10^{-3}\) to decimal form

As the exponent is -3, we will move the decimal point 3 places to the left: \(7.87 \times 10^{-3} = 0.00787\)

(g) Convert \(9.87 \times 10^{7}\) to decimal form

As the exponent is 7, we will move the decimal point 7 places to the right: \(9.87 \times 10^7 = 98,700,000\)

(h) Convert \(3.7899 \times 10^{2}\) to decimal form

As the exponent is 2, we will move the decimal point 2 places to the right: \(3.7899 \times 10^2 = 378.99\)

(i) Convert \(1.093 \times 10^{-1}\) to decimal form

As the exponent is -1, we will move the decimal point 1 place to the left: \(1.093 \times 10^{-1} = 0.1093\)

(j) Convert \(2.9004 \times 10^{\circ}\) to decimal form

There seems to be a mistake in the given expression, as the exponent should be a number, not a symbol. Please check the expression and provide the correct one.

(k) Convert \(3.9 \times 10^{-4}\) to decimal form

As the exponent is -4, we will move the decimal point 4 places to the left: \(3.9 \times 10^{-4} = 0.00039\)

(1) Convert \(1.904 \times 10^{-8}\) to decimal form

As the exponent is -8, we will move the decimal point 8 places to the left: \(1.904 \times 10^{-8} = 0.00000001904\)

Key concepts

Decimal Conversion

Decimal conversion involves transforming a number expressed in scientific notation into its standard decimal form. This requires moving the decimal point based on the exponent of ten in the notation. For example, with \(2.98 \times 10^{-5}\), the exponent -5 tells us to shift the decimal point five places to the left, resulting in \(0.0000298\).
When the exponent is positive, like \(4.358 \times 10^{9}\), the decimal point moves to the right, producing a larger number: \(4,358,000,000\).
In summary, a positive exponent shifts the decimal point right, increasing the number's value, while a negative exponent shifts it left, decreasing the value.

• Negative exponent: Move decimal left.
• Positive exponent: Move decimal right.

Exponent Rules

Exponent rules are essential in scientific notation. They dictate how we move the decimal point when converting to or from scientific notation. The value of the exponent signifies the number of spaces to shift the decimal point. In contrast to decimal conversion, exponent rules provide a systematic way to handle very large or small numbers.
For instance, an expression with \(10^{-3}\) means moving the decimal three places left. This helps in minimizing the number of zeros we write or read. Alternatively, \(10^{2}\) directs us to shift two places to the right, expanding the number.
By understanding these rules, we simplify both the notation process and our interaction with large data sets in mathematical computations.

• Exponent 0 means no shift in decimal point.
• Exponent -n means moving left by n spaces.
• Exponent +n means moving right by n spaces.

Significant Figures

Significant figures reflect the precision of a number in mathematics and science. They encompass all non-zero digits, digits between non-zero numbers, and trailing zeros in a decimal.
When dealing with numbers in scientific notation, significant figures become more apparent. For example, in \(9.87 \times 10^{7}\), each digit contributes to the precision. The number \(98,700,000\) retains its precision owing to these significant figures.
Understanding significant figures is fundamental in ensuring that calculations do not imply unwarranted accuracy. During decimal conversions, maintaining these figures ensures results remain reliable.
Consequently, accurate scientific and mathematical outcomes rely on respecting and preserving significant figures.

• All non-zero numbers are significant.
• Zeros between significant digits are significant.
• Trailing zeros in a decimal representation are significant.