1 Algebra II Properties of Exponents 2015-11-09 www.njctl.org 2 Table of Contents Click on topic to go to that section. Review of Integer Exponents Fractional Exponents Exponents with Multiple Terms Identifying Like Terms Evaluating Exponents Using a Calculator Standards 3 Review of Integer Exponents Return to Table of Contents 4 Powers of Integers Just as multiplication is repeated addition, an exponent represents repeated multiplication. For example, a5 read as "a to the fifth power" = a · a · a · a · a In this case, a is the base and 5 is the exponent. The base, a, is multiplied by itself 5 times. 5 Powers of Integers Make sure when you are evaluating exponents of negative numbers, you keep in mind the meaning of the exponent and the rules of multiplication. For example, , which is the same as . However, Notice the difference! Similarly, but . 6 1 Evaluate: 64 Answer 7 2 Evaluate: -128 Answer 8 3 Evaluate: 81 Answer 9 Properties of Exponents The properties of exponents follow directly from expanding them to look at the repeated multiplication they represent. Work to understand the process by which we find these properties and if you can't recall what to do, just repeat these steps to confirm the property. We'll use 3 as the base in our examples, but the properties hold for any base. We show that with base a and powers b and c. We'll use the facts that: 10 Properties of Exponents We need to develop all the properties of exponents so we can discover one of the inverse operations of raising a number to a power which is finding the root of a number. This concept will emerge from the final property we'll explore. But, getting to that property requires understanding the others first. 11 Multiplying with Exponents When multiplying numbers with the same base, add the exponents. e c i t c a r P h t a M 12 Dividing with Exponents When dividing numbers with the same base, subtract the exponent of the denominator from that of the numerator. 13 4 Simplify: A B C C Answer D 14 5 Simplify: A B B C Answer D 15 6 Simplify: A B C Answer C D 16 7 Simplify: A B C Answer C D 17 8 Simplify: A B A Answer C D 18 9 Simplify: A B B Answer C D 19 10 Simplify: A D B Answer C D 20 An Exponent of Zero Any base raised to the power of zero is equal to 1. Based on the property for multiplication: We know that any number times 1 is equal to itself, thus 21 11 Evaluate: 1 Answer 22 12 Evaluate: 2 Answer 23 13 Evaluate: 7 Answer 24 Negative Exponents A negative exponent moves the number from the numerator to denominator, and vice versa. Based on the multiplication and zero exponent properties: Dividing by 31, we see that: 25 Negative Exponents By definition: 26 14 Rewrite an equivalent to the following using positive exponents: A Answer B 27 15 Rewrite an equivalent to the following using positive exponents: A B A Answer 28 16 Rewrite an equivalent to the following using positive exponents: A B C Answer C D 29 17 Rewrite an equivalent to the following using positive exponents: A D B Answer C D 30 18 Which expression is equivalent to ? A B A Answer C D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. 31 19 What is the value of ? A B B Answer C D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. 32 20 Which expression is equivalent to ? A B C C Answer D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. 33 21 Which expressions are equivalent to ? Select all that apply. A B B & E Answer C D E F From PARCC EOY sample test non-calculator #13 34 22 Which expressions are equivalent to ? Select all that apply. A B B, D & E Answer C D E F From PARCC EOY sample test non-calculator #13 35 Raising Exponents to Higher Powers When raising a number with an exponent to a power, multiply the exponents. 36 Raising Exponents to Higher Powers It's important to note that when you have multiple numbers and variables being raised to a higher power, the exponent outside gets multiplied by EACH exponent inside. For example, 37 23 Simplify A B D Answer C D 38 24 Simplify A B Answer C D 39 25 Simplify A B D C Answer D 40 26 Simplify: A B B C Answer D 41 27 When is divided by , and , the quotient is A r e B w s n C A D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. 42 28 If and , what is the value of ? A r e B w s n C A D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. 43 29 The expression is equivalent to: A B B C Answer D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. 44 30 The expression is equivalent to: A B D C Answer D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. 45 Fractional Exponents Return to Table of Contents 46 Exponents & Roots We already know that roots can "undo" an exponent, and exponents can "undo" roots. For example, and Next, we will be introduced to a way to "undo" exponents using other exponents. In other words, we will find exponents which will operate the same as roots. 47 Roots as Exponents The square root of a number is equivalent to taking that number to the 1/2 exponent. To prove this we will begin by considering the multiplication property for exponents, And by definition, Comparing the 2 equations, we see that: And furthermore, 48 Special Reciprocal Exponents Similarly, raising a number to the power 1/3 is the same as taking the cube root of the number. e c i t c a And in a general, a number to the power 1/n is the same as r P h taking the nth root of the number. t a M The inverse of raising a number to a power is raising it to the reciprocal power. 49 Reciprocal Exponents So raising a number to the power of 1/n is the same as taking the nth root of the number. If you raise a number to n, then take the nth root, you get back where you started...the original number. It's the same as taking a number and adding and then subtracting the same amount. Or, multiplying and then dividing by the same number. The root is an inverse of raising to a power. And, raising to the power 1/n is the inverse of raising to n. 50 31 Evaluate: r e w s n A 51 32 Evaluate: r e w s n A 52 33 Evaluate: r e w s n A 53 34 Evaluate: r e w s n A 54 35 Evaluate: r e w s n A 55 36 Evaluate: r e w s n A 56 37 Evaluate: r e w s n A 57 38 Evaluate: r e w s n A 58 39 Evaluate: r e w s n A 59 40 Evaluate: r e w s n A 60 41 Evaluate: r e w s n A 61 42 Evaluate: r e w s n A 62 43 Evaluate: r e w s n A 63 44 Evaluate: r e w s n A 64 45 Evaluate: r e w s n A 65 Fractional Exponents with Numerators ≠ 1 A fractional exponent is performing two different operations. The numerator raises the base to a power, and the denominator takes the root of the base. In this case, the numerator raises a to the power m while the denominator takes the nth root. The order they are applied does not matter. They are commutative. 66 Power Over Root s e t o N r e h c solar POWER a e T tree ROOTS 67 Exponents with Numerators ≠ 1 While order doesn't affect the result, it's often easier to do the root first, so the number is smaller. e c i t c 5/4 a For instance, let's evaluate (81) , both ways. r P h 5 1/4 1/4 5 t (81 ) (81 ) a M (3,486,784,401)1/4 (3)5 243 243 The result is the same, but the solution on the left would be very difficult without a calculator, while that on the right is pretty easy. Generally, taking the root first is easier. Also, there's one other property worth noting. Can you find a shortcut? How would your shortcut make the problem easier? 68 Exponents with Numerators ≠ 1 (81)5/4 By the property of the multiplication of exponents (81)4/4 (81)1/4 Since 4/4 = 1 (81)1 (81)1/4 Any number raised to the power 1 is itself (81)(81)1/4 The 4th root of 81 is 3 (81)(3) Simple multiplication 243 So, if the exponent is an improper fraction, you can save time by just separating out the numerator into an integer and the fractional remainder.