Hawkes Learning Systems: College Algebra

1. Hawkes Learning Systems: College Algebra Section 1.3b: Properties of Exponents

2. Objectives • Scientific notation. • Working with geometric formulas.

3. Scientific Notation Scientific Notation A number is in scientific notation when it is written in the form Where and n is an integer. If n is a positive number, the number is large in magnitude, and if n is a negative number, the number is small in magnitude (close to 0). The number a itself can be either positive or negative and the sign of a determines the sign of the number as a whole.

4. Example 1: Scientific Notation The distance from Earth to the sun is approximately 93,000,000 miles. Scientific notation takes advantage of the observation that multiplication of a number by 10 moves the decimal point one place to the right, and we can repeat this process as many times as necessary. Thus, in scientific notation,

5. Example 2: Scientific Notation a. The mass of an electron, in kilograms, is approximately 0.000000000000000000000000000000911. Scientific notation takes advantage of the observation that multiplication of a number by moves the decimal point one place to the left, and we can repeat this process as many times as necessary. Thus, in scientific notation,

6. Example 2: Scientific Notation (cont.) b. The speed of light is 300,000,000 meters/second. What is this number in scientific notation?

7. Example 3: Scientific Notation Simplify the following expressions. a. Note: This is in scientific notation, and is an appropriate answer.

8. Example 3: Scientific Notation (cont.) b. Note: The standard notation form of this answer is very large, so we will leave it in scientific notation.

9. Working with Geometric Formulas Exponents occur in a very natural way within geometric formulas. Often, the geometric formulas you may reference in the inside front cover of the text book can be derived from simpler formulas. We will look at several examples of how to build geometric formulas from known formulas.

10. Example 4: Working with Geometric Formulas • A box with six rectangular faces is characterized by its length (l), its width (w) and its height (h). The area of a rectangle is l × w. The formula for the surface area of a box, then, is the sum of the areas of the six sides. If we let S stand for the total surface area, we obtain the formula or .

11. Example 4: Working with Geometric Formulas (cont.) The volume of a sphere of radius r is . A birdbath in the shape of a half sphere will have half of the volume of a sphere. Let V stand for the volume of the birdbath.

12. Example 4: Working with Geometric Formulas (cont.) c. A soup can is characterized by its height h and the radius r of the circles that make up the base and the top. To determine the surface area, imagine removing the top and bottom of the can and flattening out the piece in the middle to make a rectangle with height h and width (this width is found by observing that the width of the rectangle is the same as the circumference of a circle). Thus, the surface area of the curved side is . The area of a circle is , so if we let S stand for the surface area of the entire can, we have h

13. Example 4: Working with Geometric Formulas (cont.) d. The volume of the gold ingot (shown to the right) is found by observing that it is a right cylinder based on a trapezoid. The area of a trapezoid is and the ingot has length l, so its volume is b l h B