Problem 92 - College Algebra

Question

Write each number in scientific notation and use scientific notation to perform the operation(s). Express the answer in scientific notation. $$ \frac{66,000 \times 0.001}{0.003 \times 0.002} $$

Step-by-Step Solution

Answer

The answer is \(1.1 \times 10^7\).

Step 1: Convert to Scientific Notation

First, convert all the numbers into scientific notation: \n- 66,000 becomes \(6.6 \times 10^4\)\n- 0.001 becomes \(1 \times 10^{-3}\) \n- 0.003 becomes \(3 \times 10^{-3}\) \n- 0.002 becomes \(2 \times 10^{-3}\) \nSo, the given fraction becomes: \n\(\frac{(6.6 \times 10^4) \times (1 \times 10^{-3})}{(3 \times 10^{-3}) \times (2 \times 10^{-3})}\)

Step 2: Perform Multiplication for Numerator and Denominator

For the multiplication, multiply the numbers and add the exponents: \n- In the numerator, \(6.6 \times 1 = 6.6\) and \(10^4 \times 10^{-3} = 10^1\), so the numerator becomes \(6.6 \times 10^1\).\n- In the denominator, \(3 \times 2 = 6\) and \(10^{-3} \times 10^{-3} = 10^{-6}\), so the denominator becomes \(6 \times 10^{-6}\). \nTherefore, the fraction becomes \(\frac{6.6 \times 10^1}{6 \times 10^{-6}}\).

Step 3: Perform Division

To perform the division, divide the numbers and subtract the exponent of the denomiator from the exponent of the numerator to get the new exponent. \n- \(6.6 \div 6 = 1.1\)\n- \(10^1 \div 10^{-6} = 10^7\)\nTherefore, the answer is \(1.1 \times 10^7\).

Key Concepts

Numerical Operations

Numerical operations are the procedures we use to manipulate numbers, such as addition, subtraction, multiplication, and division. In the case of scientific notation, these operations include multiplying and dividing both the base numbers (the coefficients) and their accompanying powers of ten.
An important rule to remember is:

• When multiplying numbers in scientific notation, you multiply the base numbers and add the exponents.
• When dividing numbers in scientific notation, you divide the base numbers and subtract the exponents.

This process allows us to perform complex calculations more efficiently, especially when dealing with very large or very small numbers. By breaking numbers into their component parts, we simplify the calculations and ensure accuracy.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. They indicate how many times a number (the base) is used in a multiplication. In scientific notation, exponents allow us to express very large or small numbers compactly. For example, instead of writing 66,000, we use scientific notation to write it as \(6.6 \times 10^4\). Here, "4" is the exponent, indicating that 10 is multiplied by itself four times.
In our calculations:

• The exponent multiplication requires adding the exponents: \(10^4 \times 10^{-3} = 10^{4 + (-3)} = 10^1\).
• The exponent division requires subtracting the exponents: \(10^1 \div 10^{-6} = 10^{1 - (-6)} = 10^7\).

Understanding how to manipulate exponents is pivotal in scientific notation, ensuring that you can easily simplify and solve exponential expressions.

Fraction Simplification

Simplifying fractions involves reducing them to their simplest form. In the context of scientific notation, fraction simplification combines both numerical operations and exponent rules.
The original problem presents a fraction made up of numbers in scientific notation:\[\frac{(6.6 \times 10^1)}{(6 \times 10^{-6})}\]To simplify this, follow these steps:

• Divide the coefficients: \(6.6 \div 6 = 1.1\).
• Apply the exponent rule for division: subtract the denominator exponent from the numerator exponent: \(10^1 \div 10^{-6} = 10^{1 - (-6)} = 10^7\).

After simplifying, we achieve the final expression \(1.1 \times 10^7\), which is much simpler and easier to interpret. Mastery of fraction simplification aids in reducing complex mathematical expressions and drawing accurate conclusions.