Question

These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation. a) \(0.000077 \times 10^{-7}\) b) \(0.000111 \times 10^{8}\) c) \(602,000 \times 10^{18}\)

Short answer

a) \(7.7 \times 10^{-12}\), b) \(1.11 \times 10^4\), c) \(6.02 \times 10^{23}\)

Step-by-step solution

Understanding Scientific Notation

Scientific notation is a way to express numbers as a product of a coefficient (usually between 1 and 10) and a power of 10. For example, in a proper scientific notation, the number is expressed as \(a \times 10^n\), where \(1 \leq |a| < 10\). We will rewrite the given numbers accordingly.

Rewrite 0.000077

First, recognize that \(0.000077\) in scientific notation is expressed by moving the decimal place seven places to the right (since its magnitude is less than 1). This gives \(7.7 \times 10^{-5}\). Now include the factor of \(10^{-7}\): \[\begin{align*}0.000077 \times 10^{-7} &= (7.7 \times 10^{-5}) \times 10^{-7} \&= 7.7 \times 10^{-12}.\end{align*}\]

Rewrite 0.000111

For \(0.000111\), we move the decimal point four places to the right, giving us \(1.11 \times 10^{-4}\). Then, multiply it by the existing \(10^8\): \[\begin{align*}0.000111 \times 10^8 &= (1.11 \times 10^{-4}) \times 10^8 \&= 1.11 \times 10^4.\end{align*}\]

Rewrite 602,000

The integer \(602,000\) is converted to scientific notation by moving the decimal five places to the left, resulting in \(6.02 \times 10^5\). Multiply this by the existing \(10^{18}\):\[\begin{align*}602,000 \times 10^{18} &= (6.02 \times 10^5) \times 10^{18}\&= 6.02 \times 10^{23}\end{align*}\]

Final Result Summary

The rewritten numbers in proper scientific notation are:a) \(7.7 \times 10^{-12}\) b) \(1.11 \times 10^4\) c) \(6.02 \times 10^{23}\)

Key concepts

Coefficient

Scientific notation makes it easy to handle very large or very small numbers by breaking them down into two key parts: the coefficient and the power of 10. The coefficient is a crucial part of scientific notation. It is the number that appears before the multiplication sign. In proper scientific notation, the coefficient should be a number between 1 and 10. This constraint means the coefficient can even be equal to 1, but never equal to 10 or more.

• The coefficient represents the significant digits of the number.
• It should include all the non-zero digits of the number, often rounded for simplicity.
• When you adjust the number into scientific notation, you adjust the decimal point of the coefficient until the number falls within this range (1 to 10).

A key skill in writing numbers in scientific notation is moving the decimal point to turn the whole number into an appropriate coefficient.

Power of 10

The power of 10 is the second essential part of scientific notation. Once you have identified the coefficient, the power of 10 determines how many places the decimal point has been moved to convert the number to this coefficient.

• If the number is greater than 1, the power of 10 is positive, indicating the number of places to the right the decimal was moved.
• If the number is less than 1, the power of 10 is negative, indicating the number of places to the left.

This concept captures the immense or minuscule scale of the original number. For example, transforming numbers like 602,000 or 0.00077 into scientific notation illustrates how the power of 10 simplifies the expression of such numbers. Converting involves shifting the decimal point, which is represented by adding or subtracting the exponent. It's essential to match this value correctly to ensure the number retains its original value when expanded.

Numbers in Scientific Notation

Combining the coefficient and the power of 10 gives us the complete scientific notation format, which is \[a \times 10^n\]where \(a\) is the coefficient and \(n\) is the power of 10. This format is a standardized way to easily and concisely represent numbers.

• Step 1 in converting is to adjust the original number into a coefficient between 1 and 10.
• Step 2 is to count how many places the decimal was moved to decide the exponent for the power of 10.

For example:- The number 0.000077 initially shifts seven places to become 7.7, giving a power of -12 when combined with an initial multiplier of \(10^{-7}\).- The number 602,000 becomes 6.02, accounted through five decimal shifts, resulting in a power of 23 when combined with \(10^{18}\).Scientific notation makes these large numbers easier to write and understand, assisting not only in calculations but also in scientific communication.