Problem 58 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by hand. Write your result in scientific notation and standard form. $$ \frac{8.2 \times 10^{2}}{2 \times 10^{-2}} $$

Step-by-Step Solution

Answer

The result is \(4.1 \times 10^4\) or 41,000.

Step 1: Set Up the Division

The expression given is \( \frac{8.2 \times 10^2 }{2 \times 10^{-2}} \). We can separate this into two parts: the division of the coefficients \( \frac{8.2}{2} \) and the division of the powers of ten \( \frac{10^2}{10^{-2}} \).

Step 2: Divide the Coefficients

Divide the numerical coefficients: \( \frac{8.2}{2} = 4.1 \). This is a straightforward division yielding 4.1.

Step 3: Apply the Laws of Exponents

When dividing powers of ten, subtract the exponents: \( 10^{2 - (-2)} = 10^{2 + 2} = 10^4 \).

Step 4: Combine the Results

Multiply the result from Step 2 and Step 3: \( 4.1 \times 10^4 \). This is the expression in scientific notation.

Step 5: Convert to Standard Form

Convert \( 4.1 \times 10^4 \) to standard form by moving the decimal four places to the right: 41000.

Key Concepts

Law of Exponents

Understanding the Law of Exponents is like unlocking a secret code that simplifies complex expressions. When dealing with expressions like the one in this problem, the laws of exponents come in handy. Here's the key rule to remember: when dividing like bases, you subtract the exponents.
For instance, with the expression \( \frac{10^2}{10^{-2}} \), we don't need to get tangled in complex arithmetic. Instead, we apply the exponent subtraction rule: \( 10^{2 - (-2)} \). Remember that subtracting a negative is the same as adding. So, it becomes \( 10^{2+2} \), which simplifies to \( 10^4 \).
These laws make it easy to manipulate and simplify numbers in exponential form, so that we can handle them with ease in scientific calculations. Keeping a firm grip on these rules can help you tackle complicated-looking problems effortlessly.

Coefficient Division

When diving into coefficient division, the task may seem daunting, but it is quite straightforward. You simply divide the numbers in front of the powers of ten. These numbers are known as coefficients.
In the original exercise, you see \( \frac{8.2}{2} \). This involves dividing 8.2 by 2. Think of it like a simple ratio: how many times does 2 fit into 8.2?

• It's like cutting a piece of string into equal parts.
• The answer is 4.1, without much fuss.

It’s about basic division skills. The easier you find it to divide these coefficients, the smoother your scientific notation calculations will be. Keep practicing these simple divisions to build your confidence in tackling more complex calculations.

Standard Form Conversion

Moving from scientific notation to standard form is a matter of shifting the decimal point. It's kind of like moving chess pieces on a board, but more predictable!
For instance, converting \( 4.1 \times 10^4 \) to standard form requires us to understand what the exponent tells us. Here, \( 10^4 \) means to move the decimal 4 places to the right.

• Think of it as making the number feel larger, which it truly is.
• Each move to the right multiplies the original number by ten.

After moving the decimal four places with this problem, you get 41000. It often helps to write out the moves to prevent confusion. Understanding this conversion not only aids calculations but also bridges the gap between different number representations. It's a critical skill in both math and science, making numbers tangible and less abstract.