Problem 27 - Algebra for College Students

Question

For Problems \(19-32\), write each of the following in ordinary decimal notation. For example, \((3.18)(10)^{2}=318\). \((4.3)(10)^{-1}\)

Step-by-Step Solution

Answer

0.43

Step 1: Understand Scientific Notation

The given expression \((4.3)(10)^{-1}\) uses scientific notation. Scientific notation expresses numbers as a product of a decimal number and a power of ten. Here, \(4.3\) is the decimal, and \(10^{-1}\) is the power of ten.

Step 2: Interpret the Power of Ten

The expression \((10)^{-1}\) tells us to divide by 10 once. Negative exponents in scientific notation indicate division by 10 for each negative exponent.

Step 3: Convert to Decimal Notation

To convert \(4.3 \times 10^{-1}\) into ordinary decimal notation, divide \(4.3\) by \(10\). This results in moving the decimal point in \(4.3\) one place to the left, resulting in \(0.43\).

Step 4: Write Final Answer

The ordinary decimal notation for \((4.3)(10)^{-1}\) is \(0.43\).

Key Concepts

Decimal Notation

Decimal notation is the standard way of writing numbers using the decimal system. Most of us encounter it daily without even realizing. It's simply writing numbers as we typically do, like 5, 12.34, or 0.678. This method uses digits from 0 to 9 and places value to represent numbers. Each digit's position relative to the decimal point signifies its value, whether it's tens, hundreds, tenths, etc.
For example, in the number 347.89:

• 3 is in the hundreds place.
• 4 is in the tens place.
• 7 is in the ones place.
• 8 is in the tenths place.
• 9 is in the hundredths place.

We can convert numbers in scientific notation (a number times a power of ten) to this more common format, which is exactly what we did when converting \( (4.3)(10)^{-1} \) to 0.43. Moving decimals based on exponents plays a key role in this conversion.

Negative Exponents

A negative exponent in mathematics signals a division rather than multiplication. When you see a negative exponent, for instance, \(10^{-1}\), it means you need to divide 1 by \(10^{1}\), or just 10 in this case. So, it makes 1/10 or 0.1.
Negative exponents can initially seem complex, but keeping the following tips in mind can help:

• Each "-1" moves the decimal one place to the left.
• You are essentially finding the reciprocal of the base raised to that exponent.
• Negative exponents apply the division continuously for larger values, like \(10^{-3} = 1/1000 = 0.001\).

This concept was key in our problem where \( (4.3)(10)^{-1} \) led us to divide 4.3 by 10 to achieve 0.43, effectively moving the decimal point left by one position.

Power of Ten

The power of ten is a way of expressing numbers using the number 10 raised to an exponent. This method is useful for very large or small numbers. For example:

• \(10^3\) equals 1,000, meaning the number 1 followed by three zeros.
• \(10^{-3}\) equals 0.001, pushing the decimal three places to the left.

In problems like ours, understanding the power of ten alters the position of the decimal in the number, which dramatically changes the number itself. The base 10 is convenient for its compatibility with our decimal system. That's why you'll frequently find powers of ten in engineering, science, and technology when working with literal billions or minuscule percentages.

Converting Numbers

Converting numbers, especially between the scientific notation to decimal notation, often involves understanding both the value represented by the exponent and the base. Here's a simplified conversion process:

• Identify the base number (like 4.3 in our example).
• Observe the exponent to know how the decimal place will shift (one step left for \(10^{-1}\)).
• Apply this shift to convert, thus aligning the number into ordinary decimal notation.

Each movement of the decimal point symbolizes either a multiplication or division, based on the exponent's sign. This skill is immensely useful in fields like chemistry or physics to express measurements or constants, showcasing how mere shifts in decimal point placement can significantly change a number's meaning. In our example, breaking down the conversion showed how a relatively simple shift could convert \( (4.3)(10)^{-1} \) to 0.43, rendering it in a more readily understandable form.