Question

Convert the following exponential numbers to scientific notation. (a) \(0.170 \times 10^{2}\) (b) \(0.00350 \times 10^{-1}\)

Short answer

(a) is already in scientific notation as \(1.70\), and (b) is \(3.50 \times 10^{-4}\).

Step-by-step solution

Understand Scientific Notation

Scientific notation is a method of expressing numbers as a product of a coefficient between 1 and 10, and a power of 10. The format is: \[ a \times 10^n \] where \(1 \leq a < 10\) and \(n\) is an integer.

Convert (a) to Scientific Notation

For \(0.170 \times 10^{2}\), we first express 0.170 in a form so that the decimal point is after the first non-zero digit. 0.170 becomes 1.70 when the decimal is moved two places to the right. Every move to the right decreases the power of ten by 1. Hence:\[ 0.170 \times 10^2 = 1.70 \times 10^0 = 1.70 \] So, there is no change in the exponent because \(0.170 \times 10^2\) is effectively already \(1.70\).

Convert (b) to Scientific Notation

For \(0.00350 \times 10^{-1}\), first express 0.00350 in a form where the decimal is after the first non-zero digit. Move the decimal three places to the right to get 3.50, which increases the power of ten by 3:\[ 0.00350 = 3.50 \times 10^{-3} \] Now, combine with \(10^{-1}\):\[ 0.00350 \times 10^{-1} = 3.50 \times 10^{-3} \times 10^{-1} = 3.50 \times 10^{-4}\]Thus, the scientific notation is \(3.50 \times 10^{-4}\).

Key concepts

Exponential Numbers

When dealing with very large or very small numbers, exponential notation makes it easier to understand and manage them. An exponential number is expressed as a base raised to a power. In scientific notation, the base is always 10. Exponential numbers are useful because:

• They allow for clear representation of significant figures in a number.
• They simplify calculations by focusing on the order of magnitude.

For example, in the exercise, a number like 0.00350 multiplied by a power of ten, such as \(10^{-1}\), becomes easier to handle once converted to scientific notation. It helps convey the scale of the number without intricate details of each decimal place. This format is particularly useful in scientific and engineering calculations where precision and large numerical values are common.

Coefficients

In scientific notation, the coefficient is the number found in front of the multiplication sign (×) and the power of ten. This number should always satisfy the condition of being greater than or equal to 1 but less than 10.For instance, when converting the number \(0.170\) to scientific notation, we notice that the coefficient becomes \(1.70\) after adjusting the decimal place. Here’s why the coefficient is crucial:

• It simplifies the initial number to a single digit before the decimal.
• It enables an easier comparison between numbers as it minimizes complexity.

If you think about our example \(0.00350\), after converting, the coefficient turns into \(3.50\). It's imperative to ensure this number follows the rule of scientific notation to correctly demonstrate both magnitude and precision of the original number.

Power of Ten

The power of ten illustrates how many places the decimal point is moved to transform a number into proper scientific notation, which involves shifting to a point where there’s one non-zero digit before the decimal place.
Every time you move the decimal to the right, you decrease the exponent by 1. Conversely, moving the decimal to the left increases the power by 1. In our examples:

• For \(0.170 \times 10^{2}\), two moves right result in \(1.70 \times 10^{0}\).
• For \(0.00350 \times 10^{-1}\), shifting three places right results in \(3.50 \times 10^{-4}\).

The power of ten works like an address, pinpointing exactly where the decimal should go in the original number, essentially mapping how far from a standard integer each number actually lies.

Converting Decimals

Converting decimals into scientific notation involves careful tracking of the decimal place movement. This process ensures that numbers are displayed in the correct notational form.

• Begin by identifying the first non-zero digit in the decimal.
• Move the decimal right or left to position it just after this digit.
• Adjust the power of ten based on how many places the decimal was moved.

For example, take \(0.00350\), where the first non-zero digit is 3. By moving the decimal three places right, you transform it into \(3.50\), increasing the exponent by 3 and making the scientific notation \(3.50 \times 10^{-4}\).Remember, the power of ten will also reflect whether this movement was to the right or left, as demonstrated with exponents being positive for left movements and negative for right ones. This step is crucial not only for correctly formatting numbers but also for understanding the true scale of tiny decimals or large numbers in scientific works.