Problem 28 - Algebra for College Students

Question

Write each of the following in ordinary decimal notation. For example, \((3.18)(10)^{2}=318\). \((5.2)(10)^{-2}\)

Step-by-Step Solution

Answer

The ordinary decimal notation is 0.052.

Step 1: Understand the Exponential Notation

The number is given in scientific notation format as \((5.2)(10)^{-2}\). This notation means that 5.2 will be scaled by the factor of \(10^{-2}\).

Step 2: Interpret the Exponent

An exponent of \(-2\) indicates that the decimal point in 5.2 needs to be shifted to the left by 2 places. This is because a negative exponent indicates division by a power of ten.

Step 3: Shift the Decimal Point

Start with the number 5.2. Moving the decimal point 2 places to the left turns it into 0.052.

Key Concepts

Scientific Notation

Scientific Notation is a method used to express very large or very small numbers in a compact form. It helps to simplify numbers, making calculations more manageable and reducing errors. In scientific notation, numbers are written as a product of two terms: a coefficient and a power of ten. The format is usually written as \[a \times 10^n\]where:

• \(a\) is a number greater than or equal to 1 but less than 10, referred to as the coefficient.
• \(10^n\) is the base 10 raised to an exponent \(n\), indicating how far the decimal point should move.

Here’s an example: For \(3.18 \times 10^2\), the coefficient is 3.18, and the exponent is 2. This means 3.18 is multiplied by 100 (since 10 squared equals 100), turning it into 318 when expressed in standard decimal form. Scientific notation is invaluable in fields like science and engineering where expressing precise measurements is crucial.

Decimal Conversion

Decimal Conversion involves changing numbers from scientific notation to regular decimal form. This process mainly requires knowing where to move the decimal point.
The movement direction—left or right—depends entirely on whether the exponent is positive or negative. Here's a brief guide:

• Positive Exponent: Move the decimal to the right.
• Negative Exponent: Move the decimal to the left.

For example, if you have \(5.2 \times 10^{-2}\), the exponent \(-2\) tells you to move the decimal point 2 places to the left, converting it to 0.052.
If you had a positive exponent, like \(5.2 \times 10^2\), you would move the decimal right, resulting in 520. Decimal conversion helps quickly translate scientific notation into more widely-used decimal numbers.

Negative Exponents

Negative Exponents are crucial for understanding how to convert numbers into decimals when expressed in scientific notation. In essence, a negative exponent indicates division rather than multiplication.
This is because when a number is raised to a negative power, it means you take the reciprocal of its positive power.
For example, \(10^{-2}\) signifies \(\frac{1}{10^2} = \frac{1}{100}\).

• Therefore, \( (5.2) \times 10^{-2} \) means you are dividing 5.2 by 100.
• This moves the decimal point two places left, transforming 5.2 into 0.052.

Understanding negative exponents is essential for correctly converting and calculating numbers in scientific notation, particularly when dealing with small quantities. They provide an intuitive way to comprehend the inverse scaling effect that divisions by powers of ten have.