Problem 71 - Elementary Algebra

Question

Convert the following problems from scientific form to standard form. $$ 3.88 \times 10^{-5} $$

Step-by-Step Solution

Answer

Question: Convert the given number in scientific notation $$3.88 \times 10^{-5}$$ to standard notation. Answer: $$0.0000388$$.

Step 1: Identify the exponent in the scientific notation

In the given scientific notation $$3.88 \times 10^{-5}$$, the exponent is $$-5$$.

Step 2: Move the decimal point according to the exponent

Since the exponent is a negative value (-5), we will move the decimal point 5 places to the left.

Step 3: Fill in any missing zeros

To convert to standard notation, we will add zeros to fill in any missing spaces while moving the decimal point. The number in standard notation is: $$ 0.0000388 $$ The given number in scientific notation, $$3.88 \times 10^{-5}$$, is equivalent to $$0.0000388$$ in standard notation.

Key Concepts

Scientific Notation

Scientific notation is a method of expressing numbers that are too large or too small to be conveniently written in decimal form. It is often used in science and engineering to simplify calculations and to express numbers with a great number of digits in a more comprehensible way. A number in scientific notation is written as the product of a number between 1 and 10 and a power of 10. For example, the number \(3.88 \times 10^{-5}\) is written in scientific notation.

Standard Form

Standard form, also known as decimal form, is the way we normally see and write numbers. In standard form, every digit is given its place value, which dictates its contribution to the total value of the number. When you convert a number from scientific notation to standard form, you're expanding it to see the full number without any exponents, just the digits themselves in a straightforward linear format. For example, the standard form of \(3.88 \times 10^{-5}\) is \(0.0000388\), where each zero corresponds to a power of ten from the original scientific notation.

Negative Exponents

A negative exponent indicates that the number is a fraction less than one. It can be thought of as the number of times you divide by 10. So, for instance, \(10^{-5}\) means that you divide 1 by 10 five times, which is equivalent to moving the decimal point in the number 1 five places to the left, resulting in \(0.00001\). Therefore, when we convert a number from scientific notation with a negative exponent to standard form, we move the decimal point left as many times as the exponent indicates.

Decimal Point Placement

In the context of converting scientific notation to standard form, decimal point placement is key. This process involves shifting the decimal point a certain number of places left or right to get the appropriate value. The direction and number of places you move the decimal point depend on the value of the exponent. With a negative exponent, as seen in our example \(3.88 \times 10^{-5}\), the decimal moves to the left. It is crucial to remember to add zeros where there are no digits present after moving the decimal point, to maintain the correct value of the number.