1. Powers, Roots, and Scientific Notation

2. Key Concepts If x is a real number, then xⁿ = x·x·x·x·x…. a total of n times. Example: 4⁵ = 4·4·4·4·4 = 1,024 Here are several key properties of exponents x m ·x n = x m + n so x²·x³ = x 2+3 = x⁵ x m ÷x n = x m – n so x⁶÷x⁴= x 6-4 = x²

3. Power of Zero Any number raised to the 0 (zero power) equals 1 x⁴÷x⁴= x 4-4 = x⁰ which equals 1 because any number raised to the power of zero is 1. So x⁰ = 1 when x does not equal 0

4. Negative Integer Exponents If x ≠ 0, then x² ÷ x⁵= x -3 and x² ÷ x⁵ = = = = For any number x (except zero) x -m =

5. Square Roots Squaring and finding the square root of a perfect square integer are inverse operations Example: 12² = 144 and =12 Every positive number has two square roots, a positive number and a negative number. For example = 6, but is also equal to -6, since (-6)² is 36. This lesson will focus on positive roots of numbers.

6. Scientific Notation A number in scientific notation consists of two factors. a number greater than or equal to 1, but less than 10 10 raised to a power Example: 93,000,000. has a decimal point after the last zero. Move the decimal point to the left so the base number is between 1 and 10. You moved the decimal point 7 places so the base number is 9.3. Count the number of places you moved and raise 10 to that power. In this example 10⁷. So 93,000,000 is equal to 9.3 x 10⁷.

7. Estimate a Square Root Without using a calculator, the estimated square root of a number that is not a perfect square can be found in four steps. Step 1: Find the two perfect squares the square root of the number falls between. Example: falls between 4 and 5 because 4² is 16 and 5² is 25 Make a fraction subtract the smaller perfect square from the number this is the numerator. The denominator will be the difference between the perfect squares.

8. Example 1 Powers A growing restaurant chain has 300 stores. The owners predict that by expanding into other countries, they will double the number of restaurants every 5 years for the next 20 years. If they are correct how many restaurants will the chain have 20 years from now? Step 1: Recognize that the number of stores is expected to double every 5 years. Therefore, in 20 years there will be 4 doublings. We can represent the doublings using exponents like this; 2 ∙ 2 ∙ 2 ∙ 2, or 2 4 Step 2: Use the exponent and the original number of restaurants to find the number predicted in 20 years. 300 x 2 4 = 300 x 16 = 4,800 stores in 20 years.

9. Example 2 Powers Area is equal to L x W. If the length of a rectangle is b 4 and the width is b 3. What is the area of the triangle? A = lw = b 4 ∙ b 3 When we multiply numbers or variables with exponents the base stays the same and we add the exponents. Therefore; b 4 ∙ b 3 = b 4+3 = b⁷ The area of the rectangle is b⁷.

10. Example 3 Powers Find the Product 2 3 ∙ 2 4 ∙ 4 0 2 3 = 2 x 2 x 2 = 8 2 4 = 2 x 2 x 2 x 2 = 16 So, 2 3 ∙ 2 4 = 8 x 16 = 128 or 2 3 ∙ 2 4 = 2 3+4 = 2 7 = 128 Multiply the product, 128 by the third term, 4 0 4 0 = 1, so 128 x 4 0 = 128 x 1 = 128