Question

Write each of the following numbers in standard scientific notation. a. \(8714 \times 10^{2}\) b. \(0.0006591 \times 10^{8}\) c. \(0.0231 \times 10^{4}\) d. \(0.00000001519 \times 10^{-6}\) e. \(12.94 \times 10^{-3}\) f. \(1.921 \times 10^{-4}\) \(g \cdot 1\) h. \(7,354,211 \times 10^{8}\)

Short answer

a. \(8.714 \times 10^6\) b. \(6.591 \times 10^4\) c. \(2.31 \times 10^2\) d. \(1.519 \times 10^{-13}\) e. \(1.294 \times 10^{-2}\) f. \(1.921 \times 10^{-4}\) g. \(1 \times 10^0\) h. \(7.354211 \times 10^{14}\)

Step-by-step solution

Convert a. \(8714 \times 10^{2}\) to standard scientific notation.

To convert this number to standard scientific notation, we need the number to be between 1 and 10. To do this, we will move the decimal point 4 places to the left so that it comes between 8 and 7. Now, we have \(8.714\). However, since we moved the decimal place 4 places to the left in addition to the exponent of 2, the final exponent will be \( 4 + 2 = 6\). So, the final number in standard scientific notation is \(8.714 \times 10^6\).

Convert b. \(0.0006591 \times 10^{8}\) to standard scientific notation.

We will move the decimal point 4 places to the right so it comes after 6, giving us \(6.591\). Our new exponent will be \(8-4 = 4\). The final number in standard scientific notation is \(6.591 \times 10^4\).

Convert c. \(0.0231 \times 10^{4}\) to standard scientific notation.

Move the decimal point 2 places to the right, so it comes after 2, giving us \(2.31\). Our new exponent will be \(4-2=2\). The final number in standard scientific notation is \(2.31 \times 10^2\).

Convert d. \(0.00000001519 \times 10^{-6}\) to standard scientific notation.

Move the decimal point 7 places to the right, so it comes after 1, giving us \(1.519\). Our new exponent will be \(-6-7=-13\). The final number in standard scientific notation is \(1.519 \times 10^{-13}\).

Convert e. \(12.94 \times 10^{-3}\) to standard scientific notation.

Move the decimal point 1 place to the left so it comes after 1, giving us \(1.294\). Our new exponent will be \(-3+1=-2\). The final number in standard scientific notation is \(1.294 \times 10^{-2}\).

Convert f. \(1.921 \times 10^{-4}\) to standard scientific notation.

This number is already in standard scientific notation since \(1 \leq 1.921 < 10\) and the exponent is an integer. So, the final number is \(1.921 \times 10^{-4}\).

Convert g. \(1\) to standard scientific notation.

This number is already in standard scientific notation since it can be expressed as \(1 \times 10^0\), because any number to the power of 0 is equal to 1.

Convert h. \(7,354,211 \times 10^{8}\) to standard scientific notation.

Move the decimal point 6 places to the left, so it comes after 7, giving us \(7.354211\). Our new exponent will be \(8+6=14\). The final number in standard scientific notation is \(7.354211 \times 10^{14}\).

Key concepts

Standard Scientific Notation

Standard scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. In this format, a number is represented as the product of two factors: a decimal part (with only one non-zero digit to the left of the decimal point) and an exponent part (a power of ten written as 10 raised to an exponent). This method ensures that numbers are both easy to read and compare.

For instance, if we take the number 8714, which is part of the original exercise, and convert it to standard scientific notation, we obtain \(8.714 \times 10^6\). Notice that the decimal part is between 1 and 10, and the exponent part signifies that the decimal point has been shifted 6 places to the right.

Standard scientific notation both simplifies complex arithmetic and provides a uniform approach to handling numerical data across various scientific fields. However, achieving the correct standard scientific notation requires an understanding of both decimal point placement and exponents, which we’ll explore in detail in the following sections.

Exponents

Exponents play a pivotal role in scientific notation. They indicate how many times a number, called the base, is multiplied by itself. In the context of scientific notation, the base is always 10, and the exponent signifies the number of places the decimal point has moved.

For instance, using the textbook exercise, when we convert \(8714 \times 10^2\) to \(8.714 \times 10^6\), the moving of the decimal point 4 places left changes the exponent from 2 to 6. This is because we're effectively multiplying the original number by 10 for each place that we've moved the decimal point.

An essential rule to remember is that moving the decimal point to the right turns the exponent more negative, and moving it to the left makes it more positive. Thus, understanding how to manipulate exponents is crucial for working with scientific notation.

Decimal Point Placement

The correct placement of the decimal point is vital when converting numbers into standard scientific notation. The decimal point should be placed so that there is one non-zero digit to its left, creating a number that’s between 1 and 10.

Consider another example from the exercise, \(0.0006591 \times 10^8\). To express it in standard scientific notation, the decimal point needs to be moved to the right to fall after the first non-zero digit, giving us \(6.591\). Then, we adjust the exponent to reflect this shift.

Making such moves with the decimal point simplifies numbers and helps in correctly determining the exponent. Remember, the exponent changes opposite to the direction the decimal moves to maintain the value of the number during the conversion.

Significant Figures

Significant figures are vital in representing the precision of a measurement or computation. They include all the numbers known with certainty plus one last digit, which is somewhat uncertain or estimated.

Scientific notation is closely tied to significant figures since it clearly shows which digits are significant. For instance, \(7,354,211 \times 10^8\) can be written as \(7.354211 \times 10^{14}\) in scientific notation, indicating that all seven digits are significant.

It's essential when conducting scientific experiments or calculations to keep track of significant figures to ensure precision and accuracy. Always note the amount of significant figures when converting to scientific notation, as this will influence how many digits you need to keep in the decimal part.