Problem 22 - Intermediate Algebra

Question

For Problems \(19-32\), write each of the following in ordinary decimal notation. $$ (7.631)\left(10^{4}\right) $$

Step-by-Step Solution

Answer

The decimal notation is 76310.

Step 1: Understand Scientific Notation

The problem \((7.631)\left(10^{4}\right)\) is given in scientific notation. This means the number 7.631 is multiplied by \(10^4\). Our task is to convert this into ordinary decimal notation.

Step 2: Identify the Power of 10

The expression contains \(10^4\), which signifies that we need to move the decimal point 4 places to the right. This is because the exponent is a positive 4.

Step 3: Move the Decimal Point

Starting with 7.631, move the decimal point 4 places to the right. The number will transform as follows: 7.631 becomes 7631.0, and then the decimal point moves 3 more places to the right, adding zeros as placeholders, resulting in 76310.

Key Concepts

Decimal Notation

In mathematics, decimal notation is a way of expressing numbers using digits and a decimal point. It's the typical method we use in everyday situations. Decimal notation is much more intuitive for most people, as it directly represents the base-10 system we commonly use. For example, the number 7.631 is already written in decimal form. It's a representation that uses digits and specifies a position for each digit relative to the decimal point:

• 7 is in the units place.
• 6 is in the tenths place.
• 3 is in the hundredths place.
• 1 is in the thousandths place.

When converting from scientific notation to decimal notation, the goal is to transform the number to express it fully in its everyday form. This process typically involves adjusting the number by powers of ten to change its place value.

Powers of Ten

Powers of ten are a key concept when working with scientific notation and converting it to decimal notation. When we say "powers of ten," we're referring to numbers like 10, 100, 1000, and so on. These are expressed as exponential functions such as \(10^1\), \(10^2\), \(10^3\), etc.

• When the exponent is positive, like \(10^4\), this indicates multiplication by such a power of ten, prompting the movement of the decimal point to the right.
• This is useful for making small numbers larger or expressing them more conveniently.

In our example, the power of ten is \(10^4\). In practice, this means moving the decimal four places to the right, transforming 7.631 into a much larger number, 76310.

Exponents in Algebra

Understanding exponents is crucial in algebra and scientific notation. An exponent indicates how many times a number, known as the base, is multiplied by itself. Written as \(b^n\), \(b\) is the base and \(n\) is the exponent. Let's look deeper into these components:

• In \(10^4\), 10 is the base and 4 is the exponent.
• The exponent tells us to use the number 10 as a factor four times: \(10 \times 10 \times 10 \times 10 = 10000\).

Exponents allow us to represent large numbers succinctly. When dealing with multiplication of exponential numbers in algebra, the exponent provides a powerful shortcut to understanding scale and size quickly. This concept is what enabled us to convert 7.631 \(\times 10^4\) into the complete decimal form, 76310, by leveraging the power of ten inherent in the expression.