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BELL RINGER 3/27 •SOLVE FOR X: 2X + 7 = -23 X = -15 3X – 4 = 11 X=5 OBJECTIVE • REVIEW EXPONENT RULES AND RATIONAL NUMBERS • GUIDED PRACTICE • DAILY CLASS ASSESSMENT MGSE 8.EE.3 Exponents 5 exponent 3 Power base Example: 125 53 means that 53 is the exponential form of the number 125. 53 means 3 factors of 5 or 5 x 5 x 5 The Laws of Exponents: #1: Exponential form: The exponent of a power indicates how many times the base multiplies itself. x x x x x x x x n n times n factors of x Example: 5 5 5 5 3 #2: Multiplying Powers: If you are multiplying Powers with the same base, KEEP the BASE & ADD the EXPONENTS! x x x m So, I get it! When you multiply Powers, you add the exponents! n mn 2 6 23 2 6 3 29 512 #3: Dividing Powers: When dividing Powers with the same base, KEEP the BASE & SUBTRACT the EXPONENTS! m x m n mn x x x n x So, I get it! When you divide Powers, you subtract the exponents! 6 2 62 4 2 2 2 2 16 #4: Power of a Power: If you are raising a Power to an exponent, you multiply the exponents! x n m So, when I take a Power to a power, I multiply the exponents x mn 32 (5 ) 5 3 2 5 5 #5: Product Law of Exponents: If the product of the bases is powered by the same exponent, then the result is a multiplication of individual factors of the product, each powered by the given exponent. xy So, when I take a Power of a Product, I apply the exponent to all factors of the product. n x y n n (ab) a b 2 2 2 #6: Quotient Law of Exponents: If the quotient of the bases is powered by the same exponent, then the result is both numerator and denominator , each powered by the given exponent. n x x n y y So, when I take a Power of a Quotient, I apply the exponent to all parts of the quotient. n 4 16 2 2 4 81 3 3 4 #7: Negative Law of Exponents: If the base is powered by the negative exponent, then the base becomes reciprocal with the positive exponent. So, when I have a Negative Exponent, I switch the base to its reciprocal with a Positive Exponent. Ha Ha! If the base with the negative exponent is in the denominator, it moves to the numerator to lose its negative sign! x m 1 m x 1 1 5 3 5 125 and 1 2 3 9 2 3 3 #8: Zero Law of Exponents: Any base powered by zero exponent equals one. x 1 0 So zero factors of a base equals 1. That makes sense! Every power has a coefficient of 1. 50 1 and a0 1 and (5 a ) 0 1 SOLUTIONS 2 2 1. 3 3 2 2. 2 3 a a a 5 2 5 2 3 4 a 3. 2s 4s 2 4 s 2 7 4. (3) (3) (3) 2 5. 3 s t s t 2 4 7 3 s 7 27 23 8s (3) 2 7 43 t 9 s t 5 9 7 SOLUTIONS 12 7. 8. 9. 10. s 4 s 9 3 5 3 12 8 s t 4 4 st 5 8 36a b 4 5 4a b s12 4 s 8 395 34 s 12 4 8 4 t s t 8 4 5 4 85 3 36 4 a b 9 ab SOLUTIONS 1. 3 2. a 2 5 3 4 3. 2a 2 3 4. 2 a b 2 10 3 a 12 3 2 a 5 3 2 23 8a 6 2 22 a 52b 32 2 4 a10b 6 16a10b 6 5. (3a ) 3 a 22 6. s t 6 12 2 2 2 2 4 3 23 43 s t s t 9a 4 SOLUTIONS 5 s 7. t 5 s 5 t 2 3 8. 5 34 3 9 2 3 8 2 4 2 2 8 st st s t 9. 4 2 r rt r 8 36a b 10 4 5 4 a b 5 8 2 9ab3 2 2 32 9 a b 2 81a b 2 6 SOLUTIONS 1. 2a b 1 0 2 1 3. a 5 a 5 2 7 4. s 4s 4s 5 1 5. 3x y 2 6. s t 3 4 2 4 0 4 3 x y 1 8 12 8 x 81y12 SOLUTIONS 1 2 7. x 9 2 3 8. 5 3 2 1 x 4 x 4 3 2 4 2 1 3 8 3 8 s t 2 2 2 4 4 s t 9. 4 4 s t s t 10 2 5 b 2 2 10 36a 9 a b 2 10. 4 5 81a 4 a b 2 2 Rational and Irrational Numbers Essential Question How do I distinguish between rational and irrational numbers? Make a Venn Diagram that displays the following sets of numbers: Reals, Rationals, Irrationals, Integers, Wholes, and Naturals. Reals Rationals 2 3 -3 -2.65 Integers -19 Wholes 1 0 6 Naturals 1, 2, 3... 4 Irrationals 2 Recall that rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals. 3 4 = 3.8 5 2 = 0.6 3 1.44 = 1.2 Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number. For example, 2 is not a perfect square, so 2 is irrational. Caution! A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all. Check It Out! Example 1 Rational, Irriational, Not a Real Number 1. 9 9 =3 rational 2. 3. –35.9 –35.9 is a terminating decimal. rational 81 3 rational 81 9 = =3 3 3 Additional Example 2: Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number. 4. 21 irrational 5. 0 3 rational 0 =0 3 Additional Example 2: Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number. 6. 4 0 not a real number Check It Out! Example 2 State if each number is rational, irrational, or not a real number. 7. 23 23 is a whole number that is not a perfect square. irrational 8. 9 0 not a real number Check It Out! Example 2 State if each number is rational, irrational, or not a real number. 9. 64 81 rational 8 9 8 64 = 9 81