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Exponents Objectives: 1. Using the product rule 2. Evaluating expressions raised to the zero power 3. Using the quotient rule 4. Evaluating expressions raised to the negative powers 5. Raising a power to a power 6. Converting between scientific notation and standard notation Exponents may be used to write repeated factors in an expression. 32 3 3 In this case 3 is the base and 2 is the power or exponent. This means we can say that 3 to the second power means that we are multiplying 3 by itself 2 times. y3 y y y In this case y is the base and 3 is the exponent. This means that y is raised to the third power or y is multiplied by itself 3 times. Both of these are examples of exponential expressions. These are expressions that can be multiplied, divided, added, and subtracted or raised to another power. In this module we look at multiplying, dividing and raising to powers. a 2 a 3 a a a a a a 23 a 5 This suggests the following rule of multiplying like bases. Product Rule for Exponents If m and n are positive exponents and a is a real number then, a m a n a m n In other words, when we multiply like bases we add the exponents. Example 1: Multiplying like bases a. 23 24 b. b 4b5 c. y 2 y y3 Solution: a. 23 24 234 27 b. b 4b5 b 45 b9 c. y 2 y y3 y213 b6 Example 2: Using the product rule and properties. 1 Simplify: a. 3 x 2 4 x b. 2a 3b 5ab 4 Solution: a. 3 x 2 4 x 3 4 x 2 x 12 x 21 12 x 3 b. 2a 3b 5ab 4 2 5a 3 a b b 4 10 a 31b1 4 10 a 4b5 We define the 0 power to be 1 in order that all the rules for exponents remain true if the exponent is zero. Zero Exponent If a does not equal 0 then a 0 1 Example 3: Zero exponents Evaluate: b. 50 d. x y 0 a. 50 c. 2x 0 Solution: a. 50 1 b. 50 1 c. 2 x 0 2 1 2 d. x y = 1 0 We now look at dividing like bases keeping in mind that division is the opposite of multiplication. Since when we multiplied like bases we added the exponents, dividing like bases would indicate that we should subtract the exponents. x5 x x x x x x 5 3 x 2 x 2 x x From this we define division as the following. 2 Quotient Rule for Exponents If a is a nonzero real number and n and m are integers, then am n a mn a In other words, when dividing like bases subtract the exponents Example 4: Using the quotient rule x7 15 x 3 y 6 a. 5 b. x 3x 2 y 4 Solution: x7 a. 5 x 75 x 2 x 15 x3 y 6 b. 2 4 5 x32 y 64 5 xy2 3x y When the exponent in the denominator is larger than the exponent of the numerator, applying the quotient rule will result in a negative exponent. For example, x5 7 x57 x 2 But we could also evaluate it as follows. x x5 x x x x x 1 2 x7 x x x x x x x x From this we see that 1 x 2 2 x In general we then define negative exponents to be the reciprocal of the base raised to the positive exponent. 3 Negative Exponents If a is a real number other than 0 and n is a positive integer, then 1 a n n a Example 5: Evaluating Expressions with Negative exponents Evaluate: c 3a 1 a. 32 b. 2 x 3 Solution: 1 1 2 1 c. 3a 1 a. 32 2 b. 2 x 3 3 9 x3 3a Raising a Power to a Power Sometimes it is necessary to raise exponential expressions themselves to a power. For example x2 3 x2 x2 x2 x222 x6 . Notice that if we had multiplied the exponents the result is the same x2 3 x23 x6 Power Rule for Exponents If m and n are positive integers and a is a real number , then am n amn amn Remember when raising a power to a power, keep the base the same and multiply the exponents. Example 6: Raising powers to powers Simplify: a. b3 b. 2x 4 5 3 Solution: a. b3 b35 b15 5 b. 2 x4 2 x43 8x12 3 3 4 Scientific Notation Very large and very small numbers occur frequently in nature. For example the distance from Earth to the Sun is very large whereas the diameter of a helium atom is very small. Some calculators do not operate with very large or very small numbers, so scientific notation is a shorthand way of writing these numbers so all calculators can handle them. Scientific Notation A positive number is written in scientific notation if it is written as the product of a number a, where 1 a 10 and an integer power of b of 10: a 10b The following are examples of numbers written in scientific notation. 2.3 104 1.6 103 To evaluate 2.3 104 means that we move the decimal place 4 places to the right. Thus, 2.3 10 4 23,000 . To evaluate 1.6 103 means that we move the decimal 3 places to the left. Therefore, 1.6 103 0.0016 Similarly to write a number in scientific notation we reverse the process. 120,000 1.2 105 Thus 0.000234 2.34 104 Writing a Number in Scientific Notation 1. Move the decimal point in the original number until the new number has a value between 1 and 10. 2. Count the number of decimal places the decimal point was moved in Step 1. If the decimal point was moved to the left, the count is positive. If the decimal point was moved to the right, the count is negative. 3. Write the product of the new number in Step 1 and 10 raised to an exponent equal to the count found in Step 2. 5 Example 7: Changing a number to scientific notation Write each number in scientific notation a. 63,000,000 b 0.000017 Solution: a. 63,000 ,000 6.3 10 7 b. 0.000017 1.7 105 Writing a Number in Standard Notation from Scientific Notation Move the decimal point in the number the same number of places as the exponent on 10. If the exponent is positive, move the decimal point to the right. If the exponent is negative, move the decimal point to the left. Example 7: Changing to Standard Notation Write each number in standard notation a. 5.24 106 b 1.98 103 Solution: a. 5.24 10 6 5,240 ,000 . Notice the decimal point has been moved 6 places to the right. b. 1.98 103 0.00198 Notice that the decimal has been moved 3 places to the left. Many calculators now perform calculations using scientific notation. Check in the user manual with your calculator to see if it converts and uses scientific notation. 6

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posted: | 7/26/2011 |

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