Problem 1 - Algebra for College Students

Question

For Problems \(1-18\), write each of the following in scientific notation. For example \(27800=(2.78)(10)^{4}\). 89

Step-by-Step Solution

Answer

89 in scientific notation is \(8.9 \times 10^1\).

Step 1: Identify Original Number

The original number we are given is 89.

Step 2: Place Decimal Appropriately

Place a decimal after the first non-zero digit, turning 89 into 8.9. This forms the base of the scientific notation.

Step 3: Determine the Exponent

Count the number of places the decimal point has shifted from the original number. The decimal moved 1 place to the left, so the exponent is 1.

Step 4: Express in Scientific Notation

Combine the base \(8.9\) with 10 raised to the power of the determined exponent (1), resulting in 89 being expressed as \(8.9 \times 10^1\).

Key Concepts

Place Value

Every digit in a number has a specific "place value" which determines its value based on its position. For example, in the number 89, the digit 8 is in the 'tens' place and the digit 9 is in the 'ones' place. This means the value of 8 is actually 80 (8 tens), and the value of 9 is 9 (9 ones). Understanding place value is crucial because it forms the basis for converting numbers into scientific notation. Moreover, it helps in identifying how many times a digit should be multiplied by powers of ten. Without understanding place value, it's challenging to correctly position the decimal point when dealing with large or small numbers.

Exponentiation

Exponentiation is the operation of raising one quantity, the "base," to the power of another, the "exponent." In scientific notation, this concept is used to represent numbers concisely. For example, when we express 89 as \(8.9 \times 10^1\), the \(10^1\) indicates the decimal point moved one place. Here, 10 is the base and 1 is the exponent. The result, 10 raised to the 1st power, is simply 10. Exponentiation allows for simplification of computations by enabling the representation of large numbers in a compact form. Thus, numbers that might otherwise be large and unwieldy become easy to handle through exponentiation.

Decimal Point Conversion

Decimal point conversion is the process of moving the decimal point to transform a number into its scientific notation form. In the example of converting 89, we move the decimal after the first non-zero digit, transforming the number to 8.9. This move is crucial because it forms the base of the scientific notation. The number of places the decimal point is shifted determines the exponent on the ten. Shifting to the left results in a positive exponent, while shifting to the right results in a negative one. Mastery of decimal point conversion is key to appropriately scaling numbers for scientific notation.

Base and Exponent

In scientific notation, numbers are expressed in the form \(a \times 10^n\), where \(a\) is the "base" and \(n\) is the "exponent." In the expression \(8.9 \times 10^1\), the base is 8.9, representing the significant digits of the original number. The exponent, in this case, 1, indicates the power of ten by which the base is multiplied. This power reflects the shift of the decimal place necessary to return to the original value. Ultimately, understanding the role of the base and exponent is essential for converting numbers into scientific notation efficiently and accurately. This representation makes complex arithmetic operations more manageable.