Problem 104 - Algebra A Combined Function

Question

Evaluate each expression using exponential rules. Write each result in standard form. $$ \frac{0.4 \times 10^{5}}{0.2 \times 10^{11}} $$

Step-by-Step Solution

Answer

The result in standard form is 0.000002.

Step 1: Simplify the Fraction

Start by simplifying the coefficients in the fraction: \( \frac{0.4}{0.2} = 2 \). This is done by simply dividing 0.4 by 0.2.

Step 2: Apply Exponent Rules

Use the quotient rule for exponents: \( \frac{10^5}{10^{11}} = 10^{5-11} = 10^{-6} \). This rule states that when dividing two exponents with the same base, you subtract the exponents.

Step 3: Merge Results

Combine the simplified coefficient from Step 1 and the result from Step 2: \( 2 \times 10^{-6} \).

Step 4: Convert to Standard Form

Write the expression \( 2 \times 10^{-6} \) in standard form: \( 0.000002 \). Move the decimal point 6 places to the left due to the negative exponent.

Key Concepts

Quotient Rule for Exponents

Understanding the quotient rule for exponents can be incredibly helpful when simplifying expressions that involve division of like bases. When you have an expression of the form \( \frac{a^m}{a^n} \), you can use the quotient rule, which states that you subtract the exponent of the denominator from the exponent of the numerator. In a mathematical expression, this rule looks like \( a^{m-n} \). For example:

• If you have \( \frac{10^5}{10^{11}} \), you would calculate \( 10^{5-11} = 10^{-6} \).
• This means that you take the difference of the exponents because you are dividing the like bases (in this case, 10).

This step simplifies the expression while keeping it manageable. Once you've subtracted the exponents, you'll have a much simpler expression to work with in later calculations or conversions.

Standard Form

The concept of standard form is utilized to express numbers in a way that is easy to read and comprehend, especially by placing numbers in a form that highlights their order of magnitude. When working with expressions like \( 2 \times 10^{-6} \), converting it to standard form involves representing it as a decimal. Here's how to do it:

• The negative exponent \(-6\) tells us to move the decimal point 6 places to the left from the number 2.
• In this case, \( 2.0 \) becomes \( 0.000002 \).

Seeing numbers in standard form can quickly give you insights into the magnitude of the number—in this case, a very small number close to zero. Standard form is often used in sciences and engineering because it makes very large or very small numbers much easier to work with.

Simplifying Fractions

The process of simplifying fractions can simplify complex expressions and make them easier to interpret. When you simplify a fraction, you're finding an equivalent fraction that uses smaller, more manageable numbers. Consider the fraction \( \frac{0.4}{0.2} \):

• You can simplify this by dividing the numerator by the denominator directly, \( \frac{0.4}{0.2} = 2 \).
• This result means that the fraction simplifies to the whole number 2 because you are essentially determining how many times the denominator fits into the numerator.

Simplifying fractions like this reduces the number of terms to consider when solving or evaluating, which can be especially useful in longer, more complex algebraic expressions.