Question

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

Short answer

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

Step-by-step solution

Identify the base and exponent in the E notation

The given number is already in the form of \(a \times 10^b\). We need to make sure that \(1 \leq a < 10\). In this case, \(a=978.65\) and \(b=12\).

Adjust the base to fit the E notation range

We need to adjust the base \(a=978.65\) to fit the range \(1 \leq a < 10\). We do this by moving the decimal point two places to the left, and compensating by increasing the exponent \(b\) by 2. This gives us \(a=9.7865\) and \(b=14\).

Write the final E notation form

Writing the number in the E notation form, we get the final answer as \(9.7865 \times 10^{14}\).

Key concepts

E Notation

Often encountered in scientific and technical fields, E notation is a compact way of expressing very large or very small numbers. It utilizes the base of 10 and is represented as \(a \times 10^b\), where \(a\) is a number between 1 and 10, and \(b\) is an integer. This notation is particularly handy when dealing with numbers that would otherwise be unwieldy to write out fully, like the mass of a planet or the size of an atom.

In practice, E notation keeps the data concise and the mathematics manageable. Converting to E notation helps simplify calculations and improves readability, especially when communicating in scientific contexts. This is why it’s commonly seen in calculators and software outputs where numbers are often displayed in this form to conserve space and maintain precision.

Exponents

Exponents are a mathematical concept that help express repeated multiplication of the same factor, often referred to as the base. When you see an expression like \(10^{b}\), the base is 10, and the exponent or power is \(b\). This structure denotes that the base is multiplied by itself \(b\) times.

Exponents enhance our ability to work with very large or small numbers effectively by simplifying the numerical expression. For example, in scientific notation where the format \(a \times 10^b\) is standard, the exponent \(b\) indicates how many places the decimal point in \(a\) should move.

• If \(b\) is positive, the decimal moves to the right.
• If \(b\) is negative, the decimal moves to the left.

Understanding exponents is key to working with scientific and E notation, as they indicate the scaling of the number and determine its magnitude.

Decimal Point Adjustment

Decimal point adjustment is critical in maintaining the standard form in scientific and E notation where the number \(a\) must be between 1 and 10. Adjusting the decimal point involves moving it to the left or right to fit this criterion.

For example, with the number 978.65, the aim is to adjust it to fall within the 1 to 10 range for the coefficient \(a\). In this case, you move the decimal point two places to the left to transform it into 9.7865. This ensures consistency in how numbers are represented in scientific contexts.

The adjustment of the decimal points also necessitates changing the exponent \(b\) accordingly. When you move the decimal point to the left, you increase \(b\) by the number of places moved. Conversely, moving it to the right would decrease \(b\). Proper adjustment ensures that the altered number still represents the same real-world quantity.