Problem 70 - Algebra and Trigonometry

Question

Write each number in decimal notation without the use of exponents. $$ -8.17 \times 10^{6} $$

Step-by-Step Solution

Answer

-8170000

Step 1: Identify the direction

Given the number is in scientific notation -8.17 x 10^6. The exponent of 10 is positive and equals 6, this indicates that the decimal point is moved to the right 6 places.

Step 2: Moving the decimal

Start moving the decimal point starting from immediately after the 8 in -8.17, counting six places to the right without adding any extra zeros.

Step 3: Write it out in decimal notation

After moving the decimal to the right 6 places, the number becomes -8170000. This is the equivalent of the given number in decimal notation without the use of exponents.

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of the same number. For instance, if you have the number \(10^6\), it means that 10 is multiplied by itself 6 times: \(10 \times 10 \times 10 \times 10 \times 10 \times 10\). This can be quite a mouthful! That’s why we use exponents to simplify things.
Exponents are especially useful in scientific notation, which is a method of writing very large or very small numbers in a compact form. It’s like shorthand for numbers! When the exponent value is positive, as in the case of \(10^6\), it tells us how many places the decimal point will move to the right. This positive exponent makes a smaller number larger by a factor of ten each time. Remember, a large positive exponent means a big number is just around the corner.

Exploring Decimal Notation

Decimal notation is the standard method of writing numbers that we use in everyday life. This is the notation you see on calculators or when you write the numbers with digits from 0 to 9. It's straightforward because it doesn't involve exponents.
For the number given in the exercise, \(-8.17 \times 10^6\), converting it to decimal notation involves removing the scientific notation's exponent form. So, after adjusting the decimal point, the number is simply written as \(-8170000\). This is an easy-readable format and what most of us are familiar with when handling numbers. Being comfortable with switching between scientific and decimal notation allows for accurate interpretation of a number's magnitude.

The Art of Moving the Decimal Point

Moving the decimal point correctly is key to converting numbers from scientific notation to decimal notation. Here, you start with the number \(-8.17\) and need to pay attention to the exponent's value, which is 6. This indicates the point needs to move 6 places to the right.
To do this correctly, start counting spaces from the current position of the point after the number 8.

1. First, move it one place right taking it after 81.
2. The second move takes you to 817, and so on.
3. By the sixth move, you fill in three zeros after the 817, solidifying its place as \(-8170000\).

Doing this turns a compact scientific format into a fully expanded number anyone can easily understand. Practicing this technique ensures you are converting between notations smoothly and accurately!