Problem 86 - College Algebra

Question

Perform the indicated operation and express the answer in decimal notation. $$ \frac{20 \times 10^{26}}{10 \times 10^{15}} $$

Step-by-Step Solution

Answer

The answer is \( 2 \times 10^{11} \).

Step 1: Break the Calculation into smaller parts

First, split the problem into two parts: \( \frac{20}{10} \) and \( \frac{10^{26}}{10^{15}}\).

Step 2: Simplify The Numerical Part

\( \frac{20}{10} \) simplifies to 2.

Step 3: Simplify The Exponential Part

Subtract the exponent of denominator from exponent of numerator, \( \frac{10^{26}}{10^{15}} \) equals to \( 10^{26-15} \).

Step 4: Execute the operation

Calculate the subtraction operation, \(10^{26-15} \) equals \(10^{11}\).

Step 5: Combine The Results

Combine the steps into the final answer, \(2 \times 10^{11}\).

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of a number by itself. For example, when you see something like \(10^{3}\), it means \(10 \times 10 \times 10\). This is a shorthand, making it easier to write and read large or complex numbers. Exponents have a few basic rules that make calculations much simpler:

• Multiplication of Same Base: \(a^m \times a^n = a^{m+n}\). This means when you multiply numbers with the same base, you add the exponents.
• Division of Same Base: \(\frac{a^m}{a^n} = a^{m-n}\). This means when you divide numbers with the same base, you subtract the exponents.
• Power of a Power: \((a^m)^n = a^{m \times n}\).

In the original exercise, using these rules simplifies the division of exponents, \(\frac{10^{26}}{10^{15}}\), to \(10^{26-15}=10^{11}\). As you see, the exponent rules simplify complex calculations.Understanding these principles helps you work with large numbers and scientific notation effortlessly.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. In mathematics, this often involves finding the largest number that exactly divides both the numerator and the denominator and then dividing both by that number.In our original exercise, the fraction \(\frac{20}{10}\) can be simplified. Since both 20 and 10 are divisible by 10, we divide both:

• Numerator: \(\frac{20}{10} = 2\)
• Denominator: \(\frac{10}{10} = 1\), resulting in 2 when written in simplest form.

Simplifying this fraction ensures that calculations are cleaner and more straightforward. It leaves us with simpler numbers to work with, reducing potential for error later in the problem. By understanding how to simplify fractions, you're able to transform complicated problems into much simpler ones, providing clarity and ease in math.

Converting to Decimal Notation

Decimal notation is the standard way of writing numbers. It's what we most commonly use in everyday life to express numbers. When a number is written in decimal notation, it uses base 10. To convert from scientific notation, which is a useful format for very large or very small numbers, follow these basic steps:

• Identify the power of 10. For example, in \(2 \times 10^{11}\), the power of 10 is \(11\).
• Move the decimal point in the number to the right for positive exponents (left for negative), equivalent to the exponent value. Here, you would move it 11 places.
• Fill in with zeros as necessary. \(2 \times 10^{11}\) becomes 200,000,000,000.

Understanding how to convert to decimal notation will help you interpret scientific results and present your calculations in a more conventional, accessible format. This skill is crucial for effectively communicating numerical information in science, engineering, and everyday situations where easily understood values are necessary.