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Warm Up Evaluate each expression for the given values of the variables.

Warm Up Evaluate each expression for the given values of the variables. 1. x 3 y 2 for x = –1 and y = 10 2. for x = 4 and y = (–7) Write each number as a power of the given base. –100. 3. 64; base 4. 4 3. 4. – 27; base ( – 3). (–3) 3. Objectives.

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Warm Up Evaluate each expression for the given values of the variables.

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  1. Warm Up Evaluate each expression for the given values of the variables. 1.x3y2 for x = –1 and y = 10 2. for x = 4 and y = (–7) Write each number as a power of the given base. –100 3. 64; base 4 43 4. –27; base (–3) (–3)3

  2. Objectives Evaluate expressions containing zero and integer exponents. Simplify expressions containing zero and integer exponents.

  3.  5  5  5  5 You have seen positive exponents. Recall that to simplify 32, use 3 as a factor 2 times: 32 = 3  3 = 9. But what does it mean for an exponent to be negative or 0? You can use a table and look for a pattern to figure it out. 55 54 53 52 51 50 5–1 5–2 625 3125 125 25 5

  4. When the exponent decreases by one, the value of the power is divided by 5. Continue the pattern of dividing by 5.

  5. Remember! Base x Exponent 4

  6. Notice the phrase “nonzero number” in the previous table. This is because 00 and 0 raised to a negative power are both undefined. For example, if you use the pattern given above the table with a base of 0 instead of 5, you would get 0º = . Also 0–6 would be = . Since division by 0 is undefined, neither value exists.

  7. Reading Math 2–4 is read “2 to the negative fourth power.”

  8. cup is equal to Example 1: Application One cup is 2–4 gallons. Simplify this expression.

  9. 5-3 m is equal to Check It Out! Example 1 A sand fly may have a wingspan up to 5–3 m. Simplify this expression.

  10. Example 2: Zero and Negative Exponents Simplify. A. 4–3 B. 70 Any nonzero number raised to the zero power is 1. 7º = 1 C. (–5)–4 D. –5–4

  11. Caution In (–3)–4, the base is negative because the negative sign is inside the parentheses. In –3–4 the base (3) is positive.

  12. Check It Out! Example 2 Simplify. a. 10–4 b. (–2)–4 c. (–2)–5 d. –2–5

  13. Use the definition Example 3A: Evaluating Expressions with Zero and Negative Exponents Evaluate the expression for the given value of the variables. x–2 for x = 4 Substitute 4 for x.

  14. Example 3B: Evaluating Expressions with Zero and Negative Exponents Evaluate the expression for the given values of the variables. –2a0b-4 for a = 5 and b = –3 Substitute 5 for a and –3 for b. Evaluate expressions with exponents. Write the power in the denominator as a product. Evaluate the powers in the product. Simplify.

  15. Check It Out! Example 3a Evaluate the expression for the given value of the variable. p–3 for p = 4 Substitute 4 for p. Evaluate exponent. Write the power in the denominator as a product. Evaluate the powers in the product.

  16. for a = –2 and b = 6 Check It Out! Example 3b Evaluate the expression for the given values of the variables. Substitute –2 for a and 6 for b. Evaluate expressions with exponents. Write the power in the denominator as a product. Evaluate the powers in the product. 2 Simplify.

  17. What if you have an expression with a negative exponent in a denominator, such as ? or Definition of a negative exponent. Substitute –8 for n. Simplify the exponent on the right side. An expression that contains negative or zero exponents is not considered to be simplified. Expressions should be rewritten with only positive exponents. So if a base with a negative exponent is in a denominator, it is equivalent to the same base with the opposite (positive) exponent in the numerator.

  18. B. Example 4: Simplifying Expressions with Zero and Negative Numbers Simplify. A. 7w–4

  19. and Example 4: Simplifying Expressions with Zero and Negative Numbers Simplify. C.

  20. rº = 1 and . Check It Out! Example 4 Simplify. a. 2r0m–3 b. c.

  21. Lesson Quiz: Part I 1. A square foot is 3–2 square yards. Simplify this expression. Simplify. 2. 2–6 3. (–7)–3 4. 60 1 5. –112 –121

  22. for a = 6 and b = 3 7. Lesson Quiz: Part II Evaluate each expression for the given value(s) of the variables(s). x–4 for x =10 6.

  23. Warm Up Evaluate each expression. 1. 123  1,000 2. 123  1,000 3. 0.003  100 4. 0.003  100 5. 104 6. 10–4 7. 230 123,000 0.123 0.3 0.00003 10,000 0.0001 1

  24. Objectives Evaluate and multiply by powers of 10. Convert between standard notation and scientific notation.

  25. Vocabulary scientific notation

  26. The table shows relationships between several powers of 10. Each time you divide by 10, the exponent decreases by 1 and the decimal point moves one place to the left.

  27. The table shows relationships between several powers of 10. Each time you multiply by 10, the exponent increases by 1 and the decimal point moves one place to the right.

  28. Example 1: Evaluating Powers of 10 Find the value of each power of 10. C. 109 A. 10–6 B. 104 Start with 1 and move the decimal point six places to the left. Start with 1 and move the decimal point four places to the right. Start with 1 and move the decimal point nine places to the right. 0.000001 10,000 1,000,000,000

  29. Writing Math You may need to add zeros to the right or left of a number in order to move the decimal point in that direction.

  30. Check It Out! Example 1 Find the value of each power of 10. b. 105 c. 1010 a. 10–2 Start with 1 and move the decimal point two places to the left. Start with 1 and move the decimal point five places to the right. Start with 1 and move the decimal point ten places to the right. 0.01 100,000 10,000,000,000

  31. Reading Math If you do not see a decimal point in a number, it is understood to be at the end of the number.

  32. Example 2: Writing Powers of 10 Write each number as a power of 10. B. 0.0001 C. 1,000 A. 1,000,000 The decimal point is six places to the right of 1, so the exponent is 6. The decimal point is four places to the left of 1, so the exponent is –4. The decimal point is three places to the right of 1, so the exponent is 3.

  33. Check It Out! Example 2 Write each number as a power of 10. b. 0.0001 c. 0.1 a. 100,000,000 The decimal point is eight places to the right of 1, so the exponent is 8. The decimal point is four places to the left of 1, so the exponent is –4. The decimal point is one place to the left of 1, so the exponent is –1.

  34. Multiplying by Powers of 10 You can also move the decimal point to find the value of any number multiplied by a power of 10. You start with the number rather than starting with 1.

  35. Example 3: Multiplying by Powers of 10 Find the value of each expression. A. 23.89  108 23.8 9 0 0 0 0 0 0 Move the decimal point 8 places to the right. 2,389,000,000 B. 467  10–3 Move the decimal point 3 places to the left. 4 6 7 0.467

  36. Check It Out! Example 3 Find the value of each expression. a. 853.4  105 Move the decimal point 5 places to the right. 853.4 0 0 0 0 85,340,000 b. 0.163  10–2 Move the decimal point 2 places to the left. 0.0 0163 0.00163

  37. Scientific notation is a method of writing numbers that are very large or very small. A number written in scientific notation has two parts that are multiplied. The first part is a number that is greater than or equal to 1 and less than 10. The second part is a power of 10.

  38. Example 4A: Astronomy Application Saturn has a diameter of about km. Its distance from the Sun is about 1,427,000,000 km. Write Saturn’s diameter in standard form. Move the decimal point 5 places to the right. 1 2 0 0 0 0 120,000 km

  39. Example 4B: Astronomy Application Saturn has a diameter of about km. Its distance from the Sun is about 1,427,000,000 km. Write Saturn’s distance from the Sun in scientific notation. Count the number of places you need to move the decimal point to get a number between 1 and 10. 1,427,000,000 1,4 2 7,0 0 0,0 0 0 9 places Use that number as the exponent of 10. 1.427  109 km

  40. Reading Math Standard form refers to the usual way that numbers are written—not in scientific notation.

  41. Check It Out! Example 4a Use the information above to write Jupiter’s diameter in scientific notation. Count the number of places you need to move the decimal point to get a number between 1 and 10. 143,000 km 1 4 3 0 0 0 5 places Use that number as the exponent of 10. 1.43  105 km

  42. Check It Out! Example 4b Use the information above to write Jupiter’s orbital speed in standard form. 1 3 0 0 0 Move the decimal point 4 places to the right. 13,000 m/s

  43. Example 5: Comparing and Ordering Numbers in Scientific Notation Order the list of numbers from least to greatest. Step 1 List the numbers in order by powers of 10. Step 2 Order the numbers that have the same power of 10

  44. Check It Out! Example 5 Order the list of numbers from least to greatest. Step 1 List the numbers in order by powers of 10. 2  10-12, 4  10-3, 5.2  10-3, 3  1014, 4.5  1014, 4.5  1030 Step 2 Order the numbers that have the same power of 10

  45. Lesson Quiz: Part I Find the value of each expression. 1. 2. 3. The Pacific Ocean has an area of about 6.4 х 107 square miles. Its volume is about 170,000,000 cubic miles. a. Write the area of the Pacific Ocean in standard 3,745,000 0.00293 form. b. Write the volume of the Pacific Ocean in scientific notation. 1.7  108 mi3

  46. Lesson Quiz: Part II Find the value of each expression. 4. Order the list of numbers from least to greatest

  47. Multiplication Properties of Exponents 7-3 Warm Up Lesson Presentation Lesson Quiz Holt Algebra 1

  48. Warm Up Write each expression using an exponent. 1.2 • 2 • 2 2.x • x • x • x 3. Write each expression without using an exponent. 4. 43 5. y2 6. m–4 23 4 • 4 • 4 y • y

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