Problem 25 - Algebra 1

Question

SCIENTIFIC NOTATION Rewrite in scientific notation. $$ 0.05 $$

Step-by-Step Solution

Answer

The scientific notation of 0.05 is \(5 \times 10^{-2}\).

Step 1: Identify 'a' and 'n'

Here, we are given a number '0.05'. This number will be used to identify 'a' and 'n'. In this instance, we will move the decimal point to the right to make 'a' between 1 and 10, which will change the number to '5'. The number of spaces that the decimal point is moved will determine 'n'.

Step 2: Find 'n'

In Step 1, we moved the decimal point 2 places to the right. Thus, to counter this, 'n' will be -2 because moving the decimal point to the right signifies a negative exponent in scientific notation.

Step 3: Formulate the Scientific Notation

Now we will apply these values into our scientific notation formula. 'a' becomes '5' and 'n' becomes '-2'. This brings us to the scientific notation expression \(5 \times 10^{-2}\).

Key Concepts

Decimal Movement

Understanding how to move the decimal point is key when you're converting numbers to scientific notation. In scientific notation, we aim to express numbers in the form of \( a \times 10^n \), where \( a \) is a number between 1 and 10. To achieve this, you may need to shift the decimal point. Here’s how it works:

• If the original number is less than 1, like 0.05, you move the decimal point to the right to make the number larger.
• The number of places you move the decimal point becomes part of the exponent \( n \). More on that later!

In our example, 0.05, the decimal point is moved two places to the right making the number 5, which is now between 1 and 10. Every move counts, literally, in scientific notation!

Negative Exponents

Negative exponents might seem daunting, but they aren't that bad once you get the hang of it. These come into play when dealing with numbers less than 1, like fractions or small decimals.

Why Negative?

When you move the decimal point to the right to make the original number fall within the 1 to 10 range, the exponent \( n \) becomes negative. This is because you're effectively dividing by ten to compensate for each step moved right.

• For example, 0.05 became 5, requiring a move of 2 places. So \( n = -2 \).

Always remember, the further you move the decimal to the right, the more negative your exponent becomes. It represents the number being a very small fraction of 1.

Formulating Scientific Notation

Bringing it all together, scientific notation streamlines large or small numbers into a format that's easier to manage in mathematical calculations.

The Formula

Scientific notation uses the form \( a \times 10^n \):

• \( a \) is a coefficient that should be equal to or greater than 1 and less than 10.
• \( n \) is the exponent which signifies the number of decimal places moved.

For the number 0.05, after determining \( a = 5 \) and \( n = -2 \), the scientific notation becomes \( 5 \times 10^{-2} \). This format efficiently indicates 0.05 as five-hundredths, leveraging the power of ten, and emphasizes the tiny value effectively.