Problem 18 - Algebra 1: Concepts and Skills

Question

Tell whether you would move the decimal left or right and how many places to write the number in decimal form. $$ 6.89 \times 10^{5} $$

Step-by-Step Solution

Answer

The decimal form of \( 6.89 \times 10^{5} \) is \( 689000 \).

Step 1: Identify the direction to move the decimal point

As the exponent of 10 in the given scientific notation is positive (5), we will move the decimal point to the right.

Step 2: Determine the number of places to move

The number of places to move the decimal point is equal to the absolute value of the exponent. Here, the exponent is 5, so the decimal point will be moved 5 places to the right.

Step 3: Move the decimal point and write the number in decimal form

Starting from its current position, move the decimal point 5 places to the right. This will involve adding zeros to the end of the number. The decimal form of the given number then becomes \( 689000 \).

Step 4: Confirm the conversion

Finally, verify that the decimal form of the number (\( 689000 \)) properly corresponds to the original scientific notation (\( 6.89 \times 10^{5} \)). This can be done by attempting to convert the decimal form back into scientific notation, which should result in the original scientific notation.

Key Concepts

Decimal Point Movement

In scientific notation, the decimal point is an essential element. It helps in expressing large or small numbers more conveniently. When converting a number from scientific notation to decimal form, the position of the decimal point changes according to the exponent of 10.

Here's how it works:

• If the exponent is positive, it indicates that the decimal point must move to the right. This is the case when dealing with large numbers.
• If the exponent is negative, the decimal point moves to the left, typically used for very small numbers.

Understanding how the decimal point shifts is crucial for accurate conversions between scientific notation and standard decimal form. For instance, in the problem, the decimal point moves to the right because the exponent is positive (5). This movement leads the value of the number to increase as zeros are added behind the original digits.

Understanding Exponents

Exponents in scientific notation show how many times you multiply the base number (in this case, 10) by itself. They provide a shorthand way to express large or small quantities. Let's break it down:

• A positive exponent indicates a multiplication that increases the number's size. For example, in the number \( 10^5 \), we multiply 10 by itself 5 times, equating to 100,000.
• A negative exponent tells us to divide, leading to a smaller number. So \( 10^{-5} \) means taking 1/100,000.

Exponents make calculations easier, especially when dealing with very big or very small numbers. In scientific notation, they not only indicate the power of ten but also directly guide where the decimal point goes, which is why understanding their role is so important.

Powers of Ten

The power of ten is what makes scientific notation so practical. It simplifies the representation of decimal numbers by scaling numbers up or down.

The base of scientific notation is always 10, and the exponent shows how many times 10 should be multiplied (or divided if the exponent is negative). Here’s a simple breakdown:

• If the exponent is positive, such as in \( 10^5 \), it results in a large number through multiplication.
• If the exponent is negative, such as in \( 10^{-3} \), you end up with a small number by dividing.

Using powers of ten, we can convert any given number into a form that is much easier to read and handle. Numbers like 6,890,000 or 0.0000689 become easily comprehensible when represented as \(6.89 \times 10^5\) or \(6.89 \times 10^{-5}\). This simplification is what makes scientific notation an invaluable tool in mathematics and sciences.