Question

Identify the decimal part, the exponential part, and the exponent of the number \(9.66 \times 10^{-5}\).

Short answer

Decimal part: 9.66, Exponential part: \times 10^{-5}, Exponent: -5

Step-by-step solution

Identify the Decimal Part

The decimal part of a number in scientific notation is the number that comes before the multiplication sign. In this case, the decimal part is 9.66.

Identify the Exponential Part

The exponential part of a number in scientific notation consists of the base, which is typically 10 in mathematics, followed by the exponent, written as a superscript. Here, the exponential part is the entire expression '10^{-5}'.

Identify the Exponent

The exponent in the exponential part indicates the power to which the base (10) is raised. It can be found after the '10' and in this example, the exponent is '-5'.

Key concepts

Understanding the Decimal Part in Scientific Notation

The decimal part when referring to scientific notation is what we could call the 'face' of the number. It's the numeric portion that you would normally write down or see without any powers or exponential indications. In scientific notation, you'll always see it positioned before the multiplication sign and the exponential part. For example, in the number \(9.66 \times 10^{-5}\), the decimal part is \(9.66\). It's important to note that in valid scientific notation, this decimal part should be between 1 and 10, meaning it usually has only one non-zero digit before the decimal point.

What makes the decimal part so crucial is that it shows the significant figures of the number, which reflects the precision of the measurement or calculation. So if a student wanted to improve their grasp on scientific notation, one exercise improvement advice would be to practice identifying and writing down the decimal part of various numbers expressed in scientific notation.

The Role of the Exponential Part in Scientific Notation

Moving onto the exponential part, which is essentially the 'engine' that powers the number to its true value. In the scientific notation of \(9.66 \times 10^{-5}\), the exponential part is \(10^{-5}\). It consists of a base, which is almost always 10 in scientific notation due to our base-10 numbering system, coupled with an exponent, which in this case is \(-5\). The exponential part tells us how many places to move the decimal point to convert the number into its standard form.

In our example, because the exponent is negative, the exponential part signals that the decimal point must be moved to the left, indicating a number that is smaller than 1. If the exponent were positive, it would mean moving the decimal point to the right. To bolster understanding, students can practice by converting numbers in scientific notation to standard form and vice versa, focusing on the manipulation of the decimal point as instructed by the exponential part.

The Exponent: A Powerful Indicator in Scientific Notation

Finally, let's address the exponent, which can be seen as the 'directions' for where and how far the decimal point should move. The exponent is the small number written above and to the right of the base 10 in the exponential part. In the number \(9.66 \times 10^{-5}\), the exponent is '-5'. The negative sign here is pivotal - it indicates that the number is very small, as the decimal point will need to be shifted five places to the left to find its equivalent standard form.

Exponents can be positive or negative. A positive exponent signifies a large number, where the decimal point moves to the right, increasing the value. In contrast, a negative exponent like in our example indicates a small number, thus moving the decimal point left. Understanding the role of the exponent is key to mastering scientific notation. It's recommended for students to practice with both positive and negative exponents to gain familiarity with how they alter the scale of a number.