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WELCOME TO THE MM204 UNIT 9 SEMINAR SECTION 6.1: EXPONENTS Multiplying Exponential Terms Add the exponents. Keep base the same. Example: x2 * x5 = x2+5 = x7 Proof that this works: x2 * x5 =x*x * x*x*x*x*x = x7 MORE EXAMPLES If there are numbers in front, we multiply those together: Example: Multiply (2x5) (3x6) = (2 ∙ 3)(x5 ∙ x6) = 6x11 Example: (3x2) (4x3) = (3 ∙ 4)(x2 ∙ x3) = 12x5 The shortcut tells us to add the exponents. POWER RULE Power Rule: If we have an exponent raised to another exponent, we multiply the exponents together and keep the base the same. Example: Use the power rule of exponents to simplify: (22)3. = 22*3 = 26 = 64 Example: Use power rule of exponents to simplify: (p3)10. = p3*10 = p30 MORE POWER RULE EXAMPLES If we have more than one thing inside the parentheses, we raise everything inside to the power. Example: Simplify the expression: (2x3)5. = 25 * x3*5 Everything inside needs to be raised to the fifth power. = 25 * x15 = 32x15 Example: Simplify the expression: (4x2)3. = 43 * x2*3 = 43 * x6 = 64x6 SECTION 6.1 CONTINUED Dividing Exponents If we have the same letter (base) on top and bottom, we can combine them by subtraction. Subtract the exponents and keep the base the same. Example: = a 5 a3 a5-3 = a2 Proof that this works: a5 a3 a *a *a *a *a a *a *a = a2 DIVIDING EXPONENTS EXAMPLES Example: Use the quotient rule of exponents to simplify = 57 - 5 Shortcut. 57 55 = 52 Example: Use the quotient rule of exponents to simplify = a12 - 11 = a1 =a Shortcut. a 12 a 11 DIVIDING EXPONENTS Bottom Exponent is Bigger: If the exponent on bottom is bigger, we subtract and keep the answer on the bottom. Example: 72 75 1 7 5 2 1 Proof: 73 7 *7 7 *7 *7 *7 *7 1 73 MORE DIVISION EXAMPLES Example: Use the quotient rule of exponents to simplify 1 y 143 y 14 Since the bottom exponent is larger, we subtract on bottom. 1 y 11 Example: Use the quotient rule of exponents to simplify y3 x2 x 12 1 x 122 1 x 10 Since the bottom exponent is larger, we subtract on bottom. SAME EXPONENTS Same Exponents If our exponents are the same, the terms will cancel to 1. x5 x5 Example: = x5 - 5 = x0 =1 Subtract the exponents. SECTION 6.1 CONTINUED Quotient Raised to a Power We raise top and bottom to the power. Examples: x y x3 2a 5 3 y3 2 (2a ) 2 52 22 a 2 52 4a 2 25 SECTION 6.2 NEGATIVE EXPONENTS Negative Exponents Always move negative exponents to make them positive. If the negative exponent is on bottom, move it to the top to make it positive. If the negative exponent is on top, move it to the bottom to make it positive. Only move the term with the negative exponent. If there’s not a “bottom” (a fraction), make one! Examples: 1 y x 2 y2 1 y2 2 1 x2 3 y 2 3y 2 NEGATIVE EXPONENT EXAMPLES b8 a-4 a c 1 4 1 = b8c1 1 = b8c a4 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 3 x y 2 -2 3 x y y2 x3 y3 x2 MORE NEGATIVE EXPONENT EXAMPLES 5x y 4 5a 4 xy 2 5 5a 4 y 2 x x 4y Only move the term(s) with a negative exponent. Leave everything else alone. 2x y 2 5 2y 5 x2 4x 6 y 2z 4y 2 x 6z SECTION 6.2: SCIENTIFIC NOTATION Scientific Notation Short way to write really big or really small numbers. You know if your number is in scientific notation when: There’s only one digit to the left of the decimal. There’s * 10some power after the decimal part. Example: Write 123,780 in scientific notation. = 1.23780 * 105 I moved the decimal from the end of the number. It went five slots to the left. MORE SCIENTIFIC NOTATION EXAMPLES Example: Convert 45,678 to scientific notation. We want the decimal point to end up between the 4 and the 5. We’ll have to move the decimal point 4 places to the left, making the power of 10 a positive 4. 45,678 = 4.5678 * 104 Example: Convert 234,005,000 to scientific notation. We want the decimal point to end up between the 2 and the 3. We’ll have to move the decimal point 8 places to the left, making the power of 10 a positive 8. 234,005,000 = 2.34005 * 108 MORE SCIENTIFIC NOTATION EXAMPLES Example: Convert 0.0000082 to scientific notation. The decimal point needs to end up between the 8 and the 2. In order for the decimal point to move there, it needs to travel 6 places to the right, making the power of 10 a negative 6. 0.0000082 = 8.2 * 10-6 Example: Convert 0.000157 to scientific notation. The decimal point needs to end up between the 1 and the 5. In order for the decimal point to move there, it needs to travel 4 places to the right, making the power of 10 a negative 4. 0.000157 = 1.57 * 10-4 THANKS FOR PARTICIPATING! AIM: tamitacker Read, read, read! Email me if you have questions. The final is in MML and is due at the end of this Unit with all the other U9 Assignments.