Simplifying Expressions with Exponents and Scientific Notation

Equations, Inequalities, and Modeling

Simplifying Expressions with Exponents and Scientific Notation

This topic covers the simplification of algebraic expressions involving exponents, scientific notation, and constants such as π. These skills are foundational in College Algebra and are frequently used in mathematical modeling and problem solving.

• Scientific Notation: A way to express very large or very small numbers using powers of ten. For example, $7.74 \times 10^{-6}$.

• Exponents: The power to which a number or expression is raised. For example, $(1.1 \times 10^8)^{-5}$ means the reciprocal of $(1.1 \times 10^8)$ raised to the fifth power.

• Order of Operations: When simplifying, follow the order: parentheses, exponents, multiplication/division, addition/subtraction.

• Using Constants: In this problem, use $\pi \approx 3.14$ as instructed.

Example Problem

Simplify the following expression using a calculator and $\pi = 3.14$:

$\displaystyle \frac{(7.74 \times 10^{-6})^2 \; (1.1 \times 10^8)^{-5}}{\pi \; (4.6 \times 10^{-2})^3}$

• Step 1: Expand each term using exponent rules:

  • $(7.74 \times 10^{-6})^2 = 7.74^2 \times (10^{-6})^2 = 59.9076 \times 10^{-12}$

  • $(1.1 \times 10^8)^{-5} = (1.1)^{-5} \times (10^8)^{-5} = \frac{1}{1.1^5} \times 10^{-40}$

  • $(4.6 \times 10^{-2})^3 = 4.6^3 \times (10^{-2})^3 = 97.336 \times 10^{-6}$

• Step 2: Substitute $\pi = 3.14$ and combine all terms:

• Step 3: Calculate the numerator and denominator separately, then divide.

Final Answer: Use a calculator to compute the final value. The process demonstrates how to handle exponents and scientific notation in algebraic expressions.

Application: These techniques are essential for solving real-world problems in science and engineering, where quantities often span many orders of magnitude.