1 ELF Reviewing Exponent Laws MCB4U - Santowski

2 (A) Review of Exponent Laws product of powers: 3 4 x 3 6 product of powers: 3 4 x x 3 6 =  add exponents if bases are equal 3 4 x 3 6 =  add exponents if bases are equal quotient of powers: 3 9 ÷ 3 2 quotient of powers: 3 9 ÷ ÷ 6 2 =  subtract exponents if bases are equal 6 9 ÷ 6 2 =  subtract exponents if bases are equal power of a power: (3 2 ) 4 power of a power: (3 2 ) 4 (3 2 ) 4 = 3 2 x 4  multiply powers (3 2 ) 4 = 3 2 x 4  multiply powers power of a product: (3 x a) 5 power of a product: (3 x a) 5 (3 x a) 5 = 3 5 x a 5 = 243a 5  distribute the exponent (3 x a) 5 = 3 5 x a 5 = 243a 5  distribute the exponent power of a quotient: (a/3) 5 power of a quotient: (a/3) 5 (a/3) 5 = a 5 ÷ 3 5 = a 5 /243  distribute the exponent (a/3) 5 = a 5 ÷ 3 5 = a 5 /243  distribute the exponent

3 (B) Review of Zero & Negative Exponent Evaluate 2 5 ÷ 2 5. Evaluate 2 5 ÷ 2 5. (i) 2 5 ÷ 2 5 = 2 5 – 5 = 2 0 (i) 2 5 ÷ 2 5 = 2 5 – 5 = 2 0 OR (ii) 2 5 ÷ 2 5 = 32 ÷ 32 = 1 OR (ii) 2 5 ÷ 2 5 = 32 ÷ 32 = 1 Conclusion is that 2 0 = 1. Conclusion is that 2 0 = 1. In general then b 0 = 1 In general then b 0 = 1 Evaluate 2 3 ÷ 2 7. Evaluate 2 3 ÷ 2 7. (i) 2 3 ÷ 2 7 = 2 3 – 7 = 2 -4 (i) 2 3 ÷ 2 7 = 2 3 – 7 = 2 -4 OR (ii) 2 3 ÷ 2 7 = 8 ÷ 128 = 1/16 = 1/2 4 OR (ii) 2 3 ÷ 2 7 = 8 ÷ 128 = 1/16 = 1/2 4 Conclusion is that 2 -4 = 1/16 = 1/2 4 Conclusion is that 2 -4 = 1/16 = 1/2 4 In general then b -e = 1/b e In general then b -e = 1/b e

4 (C) Review of Rational Exponent We will use the Law of Exponents to prove that 9 ½ = % 9. We will use the Law of Exponents to prove that 9 ½ = % 9. 9 ½ x 9 ½ = 9 (½ + ½) = ½ x 9 ½ = 9 (½ + ½) = 9 1 Therefore, 9 ½ is the positive number which when multiplied by itself gives 9  The only number with this property is 3, or % 9 Therefore, 9 ½ is the positive number which when multiplied by itself gives 9  The only number with this property is 3, or % 9 So what does it mean? It means we are finding the second root of 9 So what does it mean? It means we are finding the second root of 9 We can go through the same process to develop a meaning to 27 1/3 We can go through the same process to develop a meaning to 27 1/3 27 1/3 x 27 1/3 x 27 1/3 = 27 (1/3 + 1/3 + 1/3) = /3 x 27 1/3 x 27 1/3 = 27 (1/3 + 1/3 + 1/3) = 27 1 Therefore, 27 1/3 is the positive number which when multiplied by itself three times gives 27  The only number with this property is 3, or 3 % 3 or the third root of 27 Therefore, 27 1/3 is the positive number which when multiplied by itself three times gives 27  The only number with this property is 3, or 3 % 3 or the third root of 27 In general, b 1/n = n / b, or that we are finding the nth root of b. In general, b 1/n = n / b, or that we are finding the nth root of b.

5 (D) The Rational Exponent m/n We can use our knowledge of Laws of Exponents to help us solve b m/n We can use our knowledge of Laws of Exponents to help us solve b m/n ex. Rewrite 32 3/5 making use of the Power of powers >>> (32 1/5 ) 3 ex. Rewrite 32 3/5 making use of the Power of powers >>> (32 1/5 ) 3 so it means we are looking for the 5th root of 32 which is 2 and then we cube it which is 8 so it means we are looking for the 5th root of 32 which is 2 and then we cube it which is 8 In general, b m/n = ( n / b) m In general, b m/n = ( n / b) m

6 (E) Important Numbers to Know The numbers 1,4,9,16,25,36,49,64,81,100,121,144 are important because... The numbers 1,4,9,16,25,36,49,64,81,100,121,144 are important because... Likewise, the numbers 1,8,27,64,125,216,343,512,729 are important because.... Likewise, the numbers 1,8,27,64,125,216,343,512,729 are important because.... As well, the numbers 1,16,81,256, 625 are important because..... As well, the numbers 1,16,81,256, 625 are important because.....

7 (F) Examples ex 1. Simplify the following expressions: ex 1. Simplify the following expressions: (i) (3a 2 b)(-2a 3 b 2 ) (i) (3a 2 b)(-2a 3 b 2 ) (ii) (2m 3 ) 4 (ii) (2m 3 ) 4 (iii) (-4p 3 q 2 ) 3 (iii) (-4p 3 q 2 ) 3 ex 2. Simplify (6x 5 y 3 /8y 4 ) 2 ex 2. Simplify (6x 5 y 3 /8y 4 ) 2 ex 3. Simplify (-6x -2 y)(-9x -5 y -2 ) / (3x 2 y -4 ) and express answer with positive exponents ex 3. Simplify (-6x -2 y)(-9x -5 y -2 ) / (3x 2 y -4 ) and express answer with positive exponents ex 4. Evaluate the following ex 4. Evaluate the following (i) (3/4) -2 (i) (3/4) -2 (ii) (-6) 0 / (2 -3 ) (ii) (-6) 0 / (2 -3 ) (iii) ( ) / (2 -3 ) (iii) ( ) / (2 -3 )

8 (F) Examples We will use the various laws of exponents to simplify expressions. We will use the various laws of exponents to simplify expressions. ex. 27 1/3 ex. 27 1/3 ex. ( ) ex. ( ) ex /4 ex /4 ex. Evaluate / /3 ex. Evaluate / /3 ex. Evaluate 4 1/2 + (-8) -1/ /3 ex. Evaluate 4 1/2 + (-8) -1/ /3 ex. Evaluate 3 / / 16 - (125) -4/3 ex. Evaluate 3 / / 16 - (125) -4/3 ex. Evaluate (4/9) ½ + (4/25) 3/2 ex. Evaluate (4/9) ½ + (4/25) 3/2

9 (G) Internet Links From West Texas A&M - Integral Exponents From West Texas A&M - Integral Exponents From West Texas A&M - Integral Exponents From West Texas A&M - Integral Exponents From West Texas A&M - Rational Exponents From West Texas A&M - Rational Exponents From West Texas A&M - Rational Exponents From West Texas A&M - Rational Exponents

10 (H) Homework Nelson textbook, p84 Nelson textbook, p84 Q1-10, 13,16,17 Q1-10, 13,16,17