Problem 42 - Algebra for College Students

Question

Use scientific notation and the properties of exponents to help you perform the following operations. \(\frac{(0.00063)(960,000)}{(3,200)(0.0000021)}\)

Step-by-Step Solution

Answer

The result is 90,000.

Step 1: Convert to Scientific Notation

Convert each number into scientific notation: - 0.00063 can be written as \(6.3 \times 10^{-4}\). - 960,000 can be written as \(9.6 \times 10^{5}\). - 3,200 can be written as \(3.2 \times 10^{3}\). - 0.0000021 can be written as \(2.1 \times 10^{-6}\).

Step 2: Rewrite the Expression

Replace every number in the original expression with their scientific notation equivalents:\[\frac{(6.3 \times 10^{-4})(9.6 \times 10^{5})}{(3.2 \times 10^{3})(2.1 \times 10^{-6})}\]

Step 3: Multiply the Numerator

Multiply the numbers and add the exponents in the numerator:\[(6.3 \times 9.6) \times (10^{-4} \times 10^{5}) = 60.48 \times 10^{1}\]

Step 4: Multiply the Denominator

Multiply the numbers and add the exponents in the denominator:\[(3.2 \times 2.1) \times (10^{3} \times 10^{-6}) = 6.72 \times 10^{-3}\]

Step 5: Divide the Results

Divide the results from the multiplication steps:\[\frac{60.48 \times 10^{1}}{6.72 \times 10^{-3}}\]Calculate separately the division of the numbers and the division of the powers of ten.\[\frac{60.48}{6.72} \times 10^{1 - (-3)} = 9 \times 10^{4}\]

Step 6: Simplify the Result

Multiply the factors to obtain the final result:\[9 \times 10^{4} = 90,000\]

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial when working with scientific notation and mathematical expressions. Exponents provide a way to simplify and manage large numbers by allowing multiplication and division of numbers in a more streamlined manner.
For instance, when multiplying numbers with exponents, like in Step 3, the exponents add:

• If you have the product \(a^m \times a^n\), it equals \(a^{m+n}\).
• This simplifies calculations immensely, as shown when \(10^{-4} \times 10^{5}\) becomes \(10^{1}\).

Similarly, for division, subtract the exponent of the divisor from the exponent of the dividend:

• In Step 5, with \(\frac{a^m}{a^n}\), it equals \(a^{m-n}\).
• This form of exponent subtraction clarified how \(10^{1} \div 10^{-3}\) results in \(10^{4}\).

These properties allow handling of very large or tiny numbers seamlessly, and are a fundamental aspect of algebra.

Algebraic Operations

Algebraic operations encompass processes like addition, subtraction, multiplication, and division, and using these with exponents can streamline calculations.
To start, converting numbers to scientific notation, as seen in Step 1, helps simplify operations by reducing complex figures into manageable formats.

• Multiplication, such as \(6.3 \times 9.6\), becomes straightforward when broken down.
• Division of these products is equally simplified, as demonstrated with \(\frac{60.48}{6.72}\) in Step 5.

The key is applying operations systematically:
- Multiplication involves combining coefficients and adding exponents.
- Division involves dividing coefficients and subtracting exponents.
Approaching it step-by-step, as detailed in the solution, minimizes errors and assures accuracy.

Mathematical Expressions

Mathematical expressions involve numbers, operations, and variables that need to be processed methodically for accurate results. In this exercise, the expression involves both multiplication and division managed by scientific notation.
Step 2 encodes a complex expression into a simpler format using scientific notation:

• Each number, like 0.00063 and 960,000, is converted as illustrated.
• This significantly reduces the complication of handling the original expression.

Breaking down expressions into scientific notations is crucial for evaluating them effectively:
- Every number gets assigned its exponential representation.
- These are then computationally easier and maintain precision.
By interpreting the given expression as \(\frac{(6.3 \times 10^{-4})(9.6 \times 10^{5})}{(3.2 \times 10^{3})(2.1 \times 10^{-6})}\)\, one can clearly simplify and solve without confusion, ensuring clarity in mathematical communication and computation.