Problem 60 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by hand. Write your result in scientific notation and standard form. $$ \frac{2.4 \times 10^{-5}}{4.8 \times 10^{-7}} $$

Step-by-Step Solution

Answer

The expression evaluates to \( 5.0 \times 10^1 \) in scientific notation and 50.0 in standard form.

Step 1: Divide the Coefficients

First, separate the coefficients in the expression. The expression is \( \frac{2.4}{4.8} \). Now, perform the division: \( 2.4 \div 4.8 = 0.5 \).

Step 2: Subtract the Exponents

Next, divide the powers of ten. According to the law of exponents, when you divide like bases, subtract the exponents: \( 10^{-5} \div 10^{-7} \) becomes \( 10^{-5 - (-7)} = 10^{7-5} = 10^2 \).

Step 3: Combine the Results

Combine the results from Step 1 and Step 2: \( 0.5 \times 10^2 \).

Step 4: Express in Scientific Notation

Convert \( 0.5 \times 10^2 \) to standard scientific notation. Move the decimal point in \( 0.5 \) one place to the right to form \( 5.0 \) and adjust the exponent appropriately: \( 5.0 \times 10^1 \).

Step 5: Write in Standard Form

Finally, express the number \( 5.0 \times 10^1 \) in standard form: this gives \( 50.0 \).

Key Concepts

Division of Exponents

When faced with an expression that involves dividing exponents, like in the original exercise, the key idea is to apply the concept of Division of Exponents. Simplifying expressions with exponents can be straightforward if you know the rules. In the division of powers with the same base, subtract the exponents. This means that for any non-zero number base \( a \), the rule \( \frac{a^m}{a^n} = a^{m-n} \) applies.
The original problem includes dividing \( 10^{-5} \) by \( 10^{-7} \). The law tells us we need to subtract the exponent of the divisor from the exponent of the dividend, i.e., \( -5 - (-7) \). Performing this subtraction gives \( 2 \), leaving us with \( 10^2 \).

• First, identify the bases that are the same.
• Then, subtract the exponents to find the new power of ten.

Standard Form

Standard form, often referred to as decimal notation, is the typical number representation we're all familiar with, like numbers used in everyday counting. When converting scientific notation to standard form, you perform the operation that results from the scientific notation's components. In the exercise solution, we started with \( 5.0 \times 10^1 \) in scientific notation.
To convert this to standard form, move the decimal point according to the power of ten. The exponent, \( 10^1 \), means moving the decimal one place to the right, changing \( 5.0 \) to \( 50.0 \).

• Look at the exponent to determine how many places to move the decimal point.
• A positive exponent moves the decimal to the right, increasing the number.
• A negative exponent moves the decimal to the left, reducing the number.

Laws of Exponents

The Laws of Exponents provide fundamental guidelines for handling mathematical expressions with exponents. These laws are crucial for simplifying expressions like the one given in the exercise. Let's break down some of the key laws that are relevant here:

• Multiplication of Exponents: When multiplying powers with the same base, add the exponents: \( a^m \times a^n = a^{m+n} \).
• Division of Exponents: For division, as we saw earlier, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m \times n} \).
• Zero Exponent: Any non-zero number raised to the exponent zero is 1: \( a^0 = 1 \).

In our step-by-step solution, we specifically used the division property, underlining how mathematics provides a structured approach to simplification. Understanding these rules enables precise simplification of expressions, crucial in algebra and beyond.