Problem 57 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by hand. Write your result in scientific notation and standard form. $$ \frac{6.3 \times 10^{-2}}{3 \times 10^{1}} $$

Step-by-Step Solution

Answer

Scientific notation: \( 2.1 \times 10^{-3} \); Standard form: 0.0021.

Step 1: Understand the Expression

We need to evaluate \( \frac{6.3 \times 10^{-2}}{3 \times 10^{1}} \). This means dividing the expression \( 6.3 \times 10^{-2} \) by \( 3 \times 10^{1} \).

Step 2: Divide the Numerator and Denominator

First, divide the coefficients: \( \frac{6.3}{3} = 2.1 \). Next, for the powers of ten, subtract the exponent in the denominator from the exponent in the numerator: \(-2 - 1 = -3\). This gives a result of \( 2.1 \times 10^{-3} \).

Step 3: Express in Scientific Notation

The result from Step 2, \( 2.1 \times 10^{-3} \), is already in scientific notation because it is of the form \( a \times 10^{n} \) where \( 1 \leq a < 10 \).

Step 4: Convert to Standard Form

To convert \( 2.1 \times 10^{-3} \) to standard form, move the decimal point 3 places to the left, giving \( 0.0021 \).

Key Concepts

Expression Evaluation

Evaluating an expression involves carrying out mathematical operations such as addition, subtraction, multiplication, or division, on a given set of numbers. In this task, we're considering the division of two exponential expressions. The expression we're working on is: \[ \frac{6.3 \times 10^{-2}}{3 \times 10^{1}} \]To evaluate such an expression by hand, begin by focusing on the different parts separately. You have coefficients (numbers like 6.3 and 3) and you have powers of ten (like \(10^{-2}\) and \(10^{1}\)). Always deal with these separately for clarity.

When evaluating, first, handle the coefficients. In our instance, you simply divide 6.3 by 3. This gives us 2.1. Then, handle the powers of ten by using exponent rules. This leads us to the section on division of exponents.

Division of Exponents

When you deal with powers of ten, division requires subtracting the exponents. This stems from the exponent rules which states that:

• \( \frac{a^{m}}{a^{n}} = a^{m-n} \)

Applying this rule to our challenge, where we have \[ \frac{10^{-2}}{10^{1}} \] here, you subtract the exponent in the denominator from the exponent in the numerator:\[ -2 - 1 = -3 \]Thus, our combined expression for the exponents becomes \(10^{-3}\). What started as separate ideas—coefficients and exponents—now meld into one expression:\[ 2.1 \times 10^{-3} \]

This result is both precise and concise in its evaluation. The expression is now ready to be considered in different forms, including the scientific and standard forms.

Standard Form Conversion

After simplifying the expression into a scientific notation of \(2.1 \times 10^{-3}\), the next common step is converting it to standard form to understand its scale in everyday numbers. Scientific notation is great for simplicity but might need translating for clarity in practical use.

To transition from scientific to standard form, you shift the decimal point. The exponent on 10 tells you how many places and which direction to move the decimal. For \(2.1 \times 10^{-3}\), the exponent -3 means moving the decimal 3 places to the left. Thus, moving 2.1 three places to the left becomes 0.0021. Having it in standard form, \(0.0021\), allows anyone to understand the number easily. This clarity is essential whenever you're working across different fields, where numbers can greatly vary in size.