Question

\(978.65 \times 10^{12}\) would be written in \(\mathrm{E}\) notation as ___________.

Short answer

Question: Convert the given value into E-notation: \(978.65 \times 10^{12}\). Answer: \(9.7865 \times 10^{14}\)

Step-by-step solution

Identify the value to be converted

The value we need to convert into E-notation is \(978.65 \times 10^{12}\).

Rewrite the value in a form with only one non-zero digit to the left of the decimal point

We need to rewrite the value as \(9.7865 \times 10^{x}\), where \(x\) is an integer. To do this, we can divide 978.65 by 100. So, $$ 9.7865 = \frac{978.65}{10^2} $$

Adjust the original exponent

Since we have divided 978.65 by 100, we need to adjust the original exponent \(10^{12}\) to account for this change. Since we divided by \(10^2\), we need to add 2 to the exponent 12. So, $$ 10^{12+2} = 10^{14} $$

Write the number in E-notation

Now that we have a value with only one non-zero digit to the left of the decimal point and we have adjusted the exponent, we can write the number in E-notation. Combining the value from Step 2 and the adjusted exponent from Step 3, we have: $$ 978.65 \times 10^{12} = 9.7865 \times 10^{14} $$ The number \(978.65 \times 10^{12}\) in E-notation is \(9.7865 \times 10^{14}\).

Key concepts

Exponents

An exponent is a mathematical notation that tells us how many times a number, called the base, is multiplied by itself. In the expression \(10^{12}\), 10 is the base, and 12 is the exponent. This means that we multiply 10 by itself 12 times. When dealing with large numbers, using exponents simplifies calculations and expressions. For example, instead of writing 1,000,000,000,000, we can write \(10^{12}\). Exponents follow specific rules that help in calculations.

• When multiplying powers with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).
• When dividing, you subtract the exponents: \(a^m \div a^n = a^{m-n}\).
• Any number raised to the power of zero is 1: \(a^0 = 1\).

Understanding exponents helps express very large or very small numbers in a more simple and concise form.

Decimal Notation

Decimal notation is a way of expressing numbers using the base 10 system, which employs digits from 0 to 9. Each digit in a number has a position that represents a power of 10. For instance, in the number 978.65:

• 9 is in the hundreds place, representing 9\( \times 10^2 \)
• 7 is in the tens place, representing 7\( \times 10^1 \)
• 8 is in the units place, representing 8\( \times 10^0 \)
• 6 is in the tenths place, representing 6\( \times 10^{-1} \)
• 5 is in the hundredths place, representing 5\( \times 10^{-2} \)

Decimal notation is especially useful for providing precise numbers and calculations. It is also the foundation of our everyday number system, making it intuitive and easy to understand.

Mathematical Conversion

Mathematical conversion involves transforming a number from one form to another without changing its value. In our example, converting 978.65 from standard decimal notation into scientific notation helps manage and operate with very large numbers.To convert a number into scientific notation or E-notation, you first rewrite the number so that there is only one non-zero digit to the left of the decimal. In our exercise, 978.65 becomes 9.7865 by dividing by 100, or \(10^2\). However, because we changed the decimal placement, we also adjust our exponent. Originally \(10^{12}\), the exponent increases by the power of ten we divided, becoming \(10^{14}\). This method simplifies the manipulation of numbers, especially in fields like science and engineering, where such notation is commonly used.

Significant Figures

Significant figures are the digits in a number that convey meaningful information about its precision. When writing numbers, especially in scientific contexts, it’s crucial to capture how precise a measurement is. In our exercise, the number 9.7865 has five significant figures:

• The digits 9, 7, 8, 6, and 5 are all counted as significant because they illustrate the precision of the original number 978.65.

Typically, all non-zero numbers are significant. Any zeros between significant figures or following a decimal are also counted. This is important in calculations – when multiplying or dividing, the result should reflect the same number of significant figures as the number with the smallest amount used in the calculation. Significant figures help ensure precision is maintained and accurately communicated in measurements, calculations, and conversion processes. Understanding them allows for clearer information transfer in scientific data and results.