Problem 44 - Algebra for College Students

Question

Use scientific notation and the properties of exponents to help you perform the following operations. \(\frac{(0.00016)(300)(0.028)}{0.064}\)

Step-by-Step Solution

Answer

The result is \(2.1 \times 10^{-2}\).

Step 1: Convert to Scientific Notation

Express each of the numbers in the expression using scientific notation. We have:\(0.00016 = 1.6 \times 10^{-4}\), \(300 = 3 \times 10^2\), \(0.028 = 2.8 \times 10^{-2}\), and \(0.064 = 6.4 \times 10^{-2}\).

Step 2: Set Up the Fraction

Substitute these values into the original fraction, yielding:\[\frac{(1.6 \times 10^{-4})(3 \times 10^2)(2.8 \times 10^{-2})}{6.4 \times 10^{-2}}\].

Step 3: Multiply the Numerator

Calculate the product of the coefficients and the powers of ten separately in the numerator:\(1.6 \times 3 \times 2.8 = 13.44\)and \(10^{-4} \times 10^{2} \times 10^{-2} = 10^{-4 + 2 - 2} = 10^{-4}\).So, the numerator becomes \(13.44 \times 10^{-4}\).

Step 4: Simplify the Expression

Now divide the product by the denominator:\[\frac{13.44 \times 10^{-4}}{6.4 \times 10^{-2}}\].Separate the coefficients and the powers of ten:\(\frac{13.44}{6.4} = 2.1\) and \(\frac{10^{-4}}{10^{-2}} = 10^{-4 + 2} = 10^{-2}\).Thus, the fraction simplifies to \(2.1 \times 10^{-2}\).

Key Concepts

Properties of Exponents

In mathematics, exponents are used to express repeated multiplication of a number by itself. Understanding the properties of exponents can simplify complex problems, especially in scientific notation. Here are some basic properties:

• Product of Powers: Add the exponents when multiplying two bases of the same base, i.e., \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers: Subtract the exponents when dividing two bases of the same base, i.e.,\(a^m / a^n = a^{m-n}\).
• Power of a Power: Multiply the exponents when raising a power to another power, i.e.,\((a^m)^n = a^{m \times n}\).

Learn these properties thoroughly, as they serve as foundational tools for working with scientific notation.

Multiplication of Scientific Notation

When multiplying numbers in scientific notation, break it down into manageable steps. Use the properties of exponents to guide you:

• Multiply the coefficients: If given two numbers in scientific notation \( (a \times 10^m) \) and \( (b \times 10^n) \), first calculate \( a \times b \).
• Add the exponents: Next, handle the \( 10^m \times 10^n \) part by adding exponents, e.g., \( 10^{m+n} \).

Combining these steps simplifies expression multiplication considerably. For example, \((1.6 \times 10^{-4})(3 \times 10^2) = 4.8 \times 10^{-2}\). This simplification makes handling very large or small numbers easier.

Division of Scientific Notation

Dividing numbers in scientific notation requires a similar process to multiplication, but with division rules:

• Divide the coefficients: With numbers \( (a \times 10^m) \) and \( (b \times 10^n) \), compute \( a / b \).
• Subtract the exponents: For \( 10^m / 10^n \), subtract the exponents, i.e., \( 10^{m-n} \).

This allows for maintaining a precise form for complex numbers. For instance, \( \frac{13.44 \times 10^{-4}}{6.4 \times 10^{-2}} = 2.1 \times 10^{-2} \). Following these steps makes operations on scientific notation a breeze, ensuring clarity in your calculations.

Simplifying Scientific Notation Expressions

Simplifying expressions in scientific notation involves using the principles of multiplication and division. Begin by handling the coefficients and exponents separately for clarity.

• Organize your work: Start by writing out the full expression in scientific notation.
• Perform operations: Multiply or divide the coefficients, then add or subtract exponents as guided by the operation.
• Combine and simplify: Reassemble the expression, ensuring the coefficient is within the typical scientific notation range (between 1 and 10).

Once simplified, expressions like \(2.1 \times 10^{-2}\) are more interpretable. Proper simplification makes calculations more efficient and results clearer.