Problem 59 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by hand. Write your result in scientific notation and standard form. $$ \frac{4 \times 10^{-3}}{8 \times 10^{-1}} $$

Step-by-Step Solution

Answer

The expression evaluates to \(5.0 \times 10^{-3}\) in scientific notation and 0.005 in standard form.

Step 1: Write Down the Expression

The expression to evaluate is \( \frac{4 \times 10^{-3}}{8 \times 10^{-1}} \). We need to simplify this expression first.

Step 2: Simplify the Coefficients

First, divide the coefficients: \( \frac{4}{8} = \frac{1}{2} \). The coefficient simplifies to 0.5.

Step 3: Simplify the Exponents of 10

Using the property of exponents, \( \frac{10^{-3}}{10^{-1}} = 10^{-3-(-1)} = 10^{-3+1} = 10^{-2} \).

Step 4: Combine Results

Combine the simplified coefficient and the power of ten: \( 0.5 \times 10^{-2} \).

Step 5: Express in Scientific Notation

To express 0.5 in scientific notation, move the decimal one place to the right: \( 5.0 \times 10^{-1} \), so the entire expression is \( 5.0 \times 10^{-3} \).

Step 6: Convert to Standard Form

To convert \( 5.0 \times 10^{-3} \) to standard form, move the decimal point 3 places to the left: 0.005.

Key Concepts

Exponents

When dealing with expressions involving exponents, it's important to understand how they work. Exponents indicate how many times a number, known as the base, is multiplied by itself. For instance, in the expression \( 10^{-3} \), the exponent \(-3\) tells us that 10 is divided by itself 3 times. This can be expressed as \( \frac{1}{10 \times 10 \times 10} \) or \( 0.001 \).

In the context of division, such as \( \frac{10^{-3}}{10^{-1}} \), we use the rules for dividing exponents. The rule is to subtract the exponent of the denominator from the exponent of the numerator, simplifying this to \( 10^{-2} \). This step gets us closer to simplifying the expression into a more manageable format.

Standard Form

Standard form is a way of writing numbers that are too big or too small in a simplified manner. The standard form allows us to express numbers without using the powers of ten. In many situations, it’s useful to convert numbers back from scientific notation to standard form.

To convert a number like \( 5.0 \times 10^{-3} \) to standard form, we need to count the decimal places. The exponent \(-3\) tells us to move the decimal 3 places to the left. This turns \( 5.0 \) into \( 0.005 \). Standard form helps in visualizing the actual value of the number, which can be especially helpful in everyday calculations and when comparing values.

Simplifying Expressions

Simplifying expressions is a key step in many math problems, including those involving scientific notation. The goal is to break down multifaceted expressions into simpler ones that are easier to work with and understand.

• Reduce Coefficients: Start by simplifying the numerical part of the expression. For instance, dividing \( 4 \) by \( 8 \) gives \( 0.5 \), which is the simplified coefficient.
• Simplify Exponents: Use exponent rules to combine or divide terms. For \( 10^{-3} / 10^{-1} \), subtract \(-1\) from \(-3\) to get \(-2\). This step reduces the complexity of the expression significantly.
• Combine Results: Multiply the simplified coefficient by the simplified power of ten to get a concise expression. The example \( 0.5 \times 10^{-2} \) becomes \( 5.0 \times 10^{-3} \) when adjusted to scientific notation.

These steps make the initial complex expression much easier to interpret and use in further calculations or applications.