1.3.1 laws of exponents

An exponent is a small number written aboveand to the right of another number. It represents the number of times you multiply by the other number. The larger number below the exponent that is used to multiply is called the base.The exponent and base together are called the power.Exponents 2 ^{3} = (2)(2)(2) = 8base 2 ^{3} = (2)(2)(2) = 8exponent 8 is a power of two. It is the third power of two. 2 ^{3} = 8power 10 Let's review: 10 ^{4} = (10)(10)(10)(10) = 10,000Which of these numbers is the base?Pick One:4 10,000 10 Let's review: 10 ^{4} = (10)(10)(10)(10) = 10,000Which of these numbers is the exponent?Pick One: 4 10,000 10 Let's review: 10 ^{4} = (10)(10)(10)(10) = 10,000Which of these numbers is the power?Pick One:4 10,000 10 25 10 ^{4} = 10,00010 is the base. 4 is the exponent. 10 ^{4} or 10,000 is the power.Simplify: 2 ^{5}16 32 2 2 ^{5}=(2)(2)(2)(2)(2) = 32Now that we've had that review, let's look at what happens when we multiply and divide powers with the same base or the same exponents. We'll develop some shortcuts to use so we can write complicated expressions more simply, especially when we're working with exponents. Let's look at multiplication first. #1: Multiplying Powers with the Same Base Simplify:(4 ^{5})(4^{6})= ((4)(4)(4)(4)(4)) ((4)(4)(4)(4)(4)(4))= (4)(4)(4)(4)(4)(4)(4)(4)(4)(4)(4) = 4 ^{11}Let's look at a couple more examples before we make a rule (pay close attention to the relationship between the exponents in the beginning and the single exponent at the end): (3 ^{2})(3^{13})= ((3)(3)) ((3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3))= (3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)= 3 ^{15}(5 ^{3})(5^{4})= ((5)(5)(5)) ((5)(5)(5)(5))= (5)(5)(5)(5)(5)(5)(5)= 5 ^{7}Keep the base and add the exponents Keep the base and subtract the exponents Multiply the bases and add the exponents Add the bases and keep the exponents Look at this example and pick a word-rule for a shortcut to simplify these kinds of expressions. (2 ^{5})(2^{3}) = 2^{8}(a ^{b})(a^{c}) = (a + a)^{bc}(a ^{b})(a^{c}) = (a)^{bc}Keep the base and add the exponents! What is the algebraic equation that represents this rule? The rule is: (a ^{b})(a^{c}) = a^{b+c}(a ^{b})(a^{c}) = a^{b-c}When multiplying powers with the same base: Keep the base and add the exponents! (a ^{b})(a^{c}) = a^{b+c}Simplify:(4 ^{5})÷(4^{2})It's easier to see how this works when we write the division as a fraction: = 4^{5} = (4)(4)(4)(4)(4) We can cancel two fours4 ^{2 }(4)(4) in the numerator anddenominator. = (4)(4)(4)(4)(4) = (4)(4)(4) = 4^{3} (4)(4) #2: Dividing Powers with the Same Base Let's look at a couple more examples before we make a rule (pay close attention to the relationship between the exponents in the beginning and the single exponent at the end): (3 ^{7})÷(3^{3}) = (3)(3)(3)(3)(3)(3)(3) = (3)(3)(3)(3) = 3^{4}(3)(3)(3) (5 ^{11})÷(5^{4})= (5)(5)(5)(5)(5)(5)(5)(5)(5)(5)(5)(5)(5)(5)(5) = (5)(5)(5)(5)(5)(5)(5) = 5 ^{7}Keep the base and add the exponents Keep the base and subtract the exponents Divide the bases and subtract the exponents Subtract the bases and keep the exponents Look at this example and pick a word-rule for a shortcut to simplify these kinds of expressions. (2 ^{5})÷(2^{3}) = 2^{2}Keep the base and subtract the exponents! (a ^{b})÷(a^{c}) = (a - a)^{bc}(a ^{b})÷(a^{c}) = (a)^{b÷c}What is the algebraic equation that represents this rule? The rule is: (a ^{b})÷(a^{c}) = a^{b+c}(a ^{b})÷(a^{c}) = a^{b-c}When dividing powers with the same base: Keep the base andsubtract the exponents! (a ^{b})÷(a^{c}) = a^{b-c}#3: Multiplying Powers with the Same Exponent Simplify:(5 ^{4})(6^{4})= ((5)(5)(5)(5)) ((6)(6)(6)(6))=(5)(5)(5)(5)(6)(6)(6)(6) =(5)(6)(5)(6)(5)(6)(5)(6) =((5)(6)) ^{4}=30 ^{4}Let's look at another example before we make a rule (pay close attention to the relationship between the bases in the beginning and the single base at the end): Simplify:(2
=
=(2)(2)(2)(5)(5)(5)
=(2)(5)(2)(5)(2)(5)
=((2)(5))
=10 Keep the base and add the exponents Keep the base and subtract the exponents Divide the bases and keep the exponent Multiply the bases and keep the exponent Look at this example and pick a word-rule for a shortcut to simplify these kinds of expressions. (3 ^{9})(4^{9}) = 12^{9}Multiply the bases and keep the exponent! (a ^{c})(b^{c}) = (a + b)^{2}^{c}(a ^{c})(b^{c}) = (a)^{bc}What is the algebraic equation that represents this rule? The rule is: (a ^{c})(b^{c}) = (ab)^{c}(a ^{c})(b^{c}) = a^{b+c}When multiplying powers with the same exponent: Multiply the basesand keep the exponent! (a ^{c})(b^{c}) = (ab)^{c}#4: Dividing Powers with the Same Exponent Simplify:(6 ^{5})÷(2^{5})It's easier to see how this works when we write the division as a fraction: = 6^{5} = (6)(6)(6)(6)(6) = (6) (6) (6) (6) (6)2 ^{5}^{ }(2)(2)(2)(2)(2) (2) (2) (2) (2) (2) = (6÷2) ^{5} = 3^{5}Let's look at another example before we make a rule (pay close attention to the relationship between the bases in the beginning and the single base at the end): Simplify:(20
= 5
= (20÷5) Keep the base and add the exponents Keep the base and subtract the exponents Divide the bases and keep the exponent Multiply the bases and keep the exponent Look at this example and pick a word-rule for a shortcut to simplify these kinds of expressions. (50 ^{9})÷(2^{9}) = 25^{9}Divide the bases and keep the exponent! (a ^{c})(b^{c}) = (a - b)^{2}^{c}(a ^{c})÷(b^{c}) = (a÷b)^{c}What is the algebraic equation that represents this rule? The rule is: (a ^{c})(b^{c}) = (ab)^{c}(a ^{c})(b^{c}) = a^{b+c}When dividing powers with the same exponent: Divide the bases and keep the exponent!(a ^{c})(b^{c}) = (a÷b)^{c}Simplify:(6 ^{5})^{2}= (6 ^{5})(6^{5})= ((6)(6)(6)(6)(6)) ((6)(6)(6)(6)(6))= (6)(6)(6)(6)(6)(6)(6)(6)(6)(6) =6 ^{10}#5: Power of a Power Let's look at a couple more examples before we make a rule (pay close attention to the relationship between the exponents in the beginning and the single exponent at the end): Simplify: (3 ^{2})^{7}= (3^{2})(3^{2})(3^{2})(3^{2})(3^{2})(3^{2})(3^{2})= ((3)(3)) ((3)(3)) ((3)(3)) ((3)(3)) ((3)(3)) ((3)(3)) ((3)(3)) = (3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3) = 3 ^{14}Simplify: (5 ^{3})^{4}= (5^{3})(5^{3})(5^{3})(5^{3})= ((5)(5)(5)) ((5)(5)(5)) ((5)(5)(5)) ((5)(5)(5)) = (5)(5)(5)(5)(5)(5)(5)(5)(5)(5)(5)(5)= 5 ^{12}Keep the base and multiply the exponents Keep the base and add the exponents Keep the base and subtract the exponents Multiply the bases and keep the exponent Look at this example and pick a word-rule for a shortcut to simplify these kinds of expressions. (9 ^{5})^{6} = 9^{30}Keep the bases and multiply the exponents! (a ^{c})(b^{c}) = (ab)^{c}(a ^{c})÷(b^{c}) = (a÷b)^{c}What is the algebraic equation that represents this rule? The rule is: (a ^{b})(a^{c}) = (a)^{b+c}(a ^{b})^{c} = a^{bc}Now you're ready to take the next quiz: 7.N.4 Part 2 - Applying the Laws of Exponents See if you remember what to do. Remember you can always retake the quizzes. Don't forget to email me if you have a question: awarren@portchesterschools.org GOOD JOB! THE END! |

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