Problem 61 - Elementary Algebra

Question

Write the following problems using scientific notation. $$ 0.0387 $$

Step-by-Step Solution

Answer

Question: Convert the number 0.0387 into scientific notation. Answer: 0.0387 can be written in scientific notation as $$3.87 \times 10^{-2}$$

Step 1: Identify the number between 1 and 10

First, we need to find the number between 1 and 10 that can be multiplied with a power of 10 to get the given number (0.0387). To do this, we move the decimal point to the right until there's only one non-zero digit to the left of the decimal point. In this case, the number becomes 3.87.

Step 2: Count the positions moved by the decimal point

Now, we have to count how many positions we moved the decimal point to the right to obtain the number 3.87 from 0.0387. We moved the decimal point 2 positions to the right.

Step 3: Write the number in scientific notation

The number 0.0387 in scientific notation is the product of the number we got in step 1 (3.87) and 10 raised to the power of the number of positions moved (2). But, since the original number was smaller than 1, we need to use a negative exponent. So, the scientific notation is: $$ 0.0387 = 3.87 \times 10^{-2} $$

Key Concepts

Algebra and Scientific Notation

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. Scientific notation is a way algebra helps us manage very large or very small numbers. It allows for easier comparison, computation, and comprehension of such numbers. In scientific notation, we express numbers as a product of a number between 1 and 10 and a power of 10. This is precisely what we do when converting the number 0.0387 into scientific notation. We apply algebraic thinking to simplify complex numbers into a standardized form, which is especially useful in fields like science and engineering where extremely precise values are common.

By breaking down 0.0387 into the product of 3.87 (a number between 1 and 10, adhering to the definition of scientific notation) and a power of 10, we've used the fundamentals of algebra to reformat a given number for more efficient use. This process involves identifying patterns, understanding the position of the decimal point, and knowing how to work with exponents - all of which are key concepts in algebra.

Decimal Point Movement in Scientific Notation

In the context of scientific notation, decimal point movement is critical. The distance that the decimal point moves corresponds to the exponent used in the notation. When we convert a decimal like 0.0387 into scientific notation, we shift the decimal point to create a new number that lies between 1 and 10. For 0.0387, we move the decimal point two places to the right to get 3.87.

Understanding the Decimal Shift

It's important to note that every shift to the right in the decimal converts to a negative exponent when we’re dealing with numbers less than one, as in the case of 0.0387. Conversely, for numbers greater than one, the movement to the left will result in a positive exponent. Decimal point movement is essentially a visual representation of the exponent by which 10 will be raised.

Exponents in Scientific Notation

Exponents play a crucial role in scientific notation. They indicate the number of times the base, which is 10 in scientific notation, is multiplied by itself. For a small number like 0.0387, after moving the decimal point two places to the right, we get 3.87, which is a larger number. To balance this increase, we use a negative exponent on 10, which indicates division by a power of 10;

Negative vs Positive Exponents

For our example, the exponent is '-2', symbolizing that we need to divide by 100 (or 10 raised to the power of 2) to revert back to the original number. This use of negative exponents is common with small numbers that are less than one. Conversely, positive exponents are used when the original number is greater than one and signifies multiplication. Remember, exponents reflect the number of places the decimal point has moved, negative for to the right (making the number smaller) and positive for to the left (making the number larger).