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Exponential Growth Laws of Exponents and Geometric Patterns

Exponents The ‘2’ is called the coefficient. The ‘x 4 ’ is called the power, of which ‘x’ is the base and ‘4’ is the exponent Consider the expression 2x 4 = 2(x)(x)(x)(x)

Laws of Exponents Multiplying powers with the same base: Notice that: (a 5 )(a 2 ) = (aaaaa)(aa) (a 5 )(a 2 ) = aaaaaaa (a 5 )(a 2 ) = a 7 When multiplying powers with the same base, we keep the base and add the exponents.

Laws of Exponents Dividing powers with the same base: Dividing powers with the same base: Notice that: When dividing powers with the same base, we keep the base and subtract the exponents. When multiplying powers with the same base, we keep the base and add the exponents.

Laws of Exponents Power of a power: When we have a power of a power, we keep the base and multiply the exponents. Notice that: When dividing powers with the same base, we keep the base and subtract the exponents. When multiplying powers with the same base, we keep the base and add the exponents.

Using the Laws of Exponents Simplify the following expressions using the first 3 exponent laws a)(w 3 )(w 10 )b) (4d 2 )(10d 6 )c) d)e) f) Answers: a) w 13 b) 40d 8 c) 5s 2 d) 2 11 e) s 8 t 8 f) 2 3

What does equal? But we could also use an exponent law: The value of any power with exponent 0 is 1. Zero and Negative Exponents Zero and Negative Exponents Powers of zero: When dividing powers with the same base, we keep the base and subtract the exponents.

Zero and Negative Exponents What does equal? But we could also use an exponent law: To change the sign of an exponent, change the base to its reciprocal. (To change the base to its reciprocal, change the sign of the exponent) To change the sign of an exponent, change the base to its reciprocal. (To change the base to its reciprocal, change the sign of the exponent) Negative powers: When dividing powers with the same base, we keep the base and subtract the exponents.

When multiplying powers with the same base, we keep the base and add the exponents. When dividing powers with the same base, we keep the base and subtract the exponents. The value of any power with exponent 0 is 1. To change the sign of an exponent, change the base to its reciprocal.(To change the base to its reciprocal, change the sign of the exponent) When we have a power of a power, we keep the base and multiply the exponents. The 5 Exponent Laws

Ex1. Simplify the following leaving no negative exponents (when possible). a) b)c) Answers: a) b) c) Ex2. Evaluate (which means “get the value of the expression”) a) b) c) d)e) Answers: a) b) c) d) −1e) 1 Zero and Negative Exponents

Ex 3. Evaluate each expression

Changing the base Notice that most of the exponent laws only work when there’s a common base. It is often helpful to be able to change the base of a power to match another power. Ex. Simplify All of these bases can be written as a base of 2. This doesn’t look like simplifying… …yet… Yep, that’s simpler. It is now expressed as a single base.

This expression has some nasty values! You are NOT required to know 81 -5 or (1/3) 7. BUT… All of these bases can be written as a base of 3: Ex 2. Evaluate without a calculator Changing the base

An exponential equations is one where the variable (unknown) is in the exponent. The only technique we have so far to solve such equations is getting a common base. Soon we will learn another, stronger technique; logarithms. Ex 1. Find the root(s), ie solve for x: We must recognize that these bases can be rewritten using base 3. Solving Exponential Equations Now, with two equal expressions, with equal bases, the exponents must also be equal.

Ex 2. Solve for x. Solving Exponential Equations Recognize the common base of 2. Now, with two equal expressions, with equal bases, the exponents must also be equal.

Ex 3. Find the root(s) of the equation below. Solving Exponential Equations Isolate the power with the variable. Recognize the common base of 2.

Your turn: Find the root(s) of the equations below. Solving Exponential Equations

Your turn: Solutions

Solving Exponential Equations Your turn: Solutions

Patterns (again) We have seen that patterns can be represented as equations: Linear Pattern: Quadratic Pattern: common difference at level 1 common difference at level 2 But what about a pattern like this? 3, 6, 12, 24, 48

Geometric Patterns We can quickly see that there is no common difference for this pattern… …but there is a common ratio. +3 +6 +12 +24 3 3 × 2 3 × 2 × 2 3 × 2 × 2 × 2 3 × 2 × 2 × 2 × 2 Or… 3 3 × 2 1 3 × 2 2 3 × 2 3 3 × 2 4 Remember: 2 0 = 1 × 2 0 ×2 ×2 So we can express this pattern as: 3, 6, 12, 24, 48 +3 +6 +12 3, 6, 12, 24, 48 Patterns with a common ratio are called geometric patterns.

Geometric Patterns We see from this that this geometric pattern can be represented by the equation: Let’s check: If n = 4 3 × 2 0 3 × 2 1 3 × 2 2 3 × 2 3 3 × 2 4 The 2 is the common ratio of the pattern and is the base of the power in the equation. The 3 is the first term of the pattern and is the coefficient in the equation. 3, 6, 12, 24, 48

Geometric Patterns In general, a geometric pattern can be written using the equation The pattern 5, 10, 20, 40, 80,… can be represented by the equation We can check by plugging in n = 5 where t 1 is the first term of the pattern (when n = 1) and where r is the common ratio.

Geometric Patterns Practice Find the 10 th term in each pattern: a)100, 50, 25, 12.5,… b)0.25, 1,4, 16,… c) 5, 8, 11, 14, … d) 1, −2, 4, −8, 16, …

Geometric Patterns Practice: Solutions Find the 10 th term in each pattern: a)100, 50, 25, 12.5,… a) CR = ½ t 1 = 100

Geometric Patterns Practice: Solutions Find the 10 th term in each pattern: b) 0.25, 1, 4, 16,… b) CR = 4 t 1 = 0.25

Geometric Patterns Practice: Solutions Find the 10 th term in each pattern: c) 5, 8, 11, 14, … c) CD = 3 t 1 = 5

Geometric Patterns Practice: Solutions Find the 10 th term in each pattern: d) 1, −2, 4, −8, 16, … d) CR = −2 t 1 = 1

Geometric Patterns Practice For the pattern below, which term has a value of 768? 6, 12, 24, 48, … The 8 th term has a value of 768.

Speed of exponential growth: an old Indian legend 1 000 000 The last square requires more than 18,000,000,000,000,000,000 grains of rice, which is equal to about 210 billion tons and is allegedly sufficient to cover the whole territory of India with a meter thick layer of rice. At a production rate of ten grains of rice per square inch, the above amount requires rice fields covering twice the surface area of the Earth, oceans included. http://www.singularitysymposium.com/e xponential-growth.html

Exponential growth examples A great many things in nature grow exponentially. Each of these situations can be modeled with a geometric pattern and thus an exponential equation. http://www.youtube.com/watch?v=gEwzDydciWc

Let’s set up a table to analyze the pattern. Number of hours since the start061218 Number of bacteria present1248 We see that the CR of this pattern is 2. Ex. A certain type of bacteria doubles every six hours. The experiment begins with 1 bacteria. However, this pattern involves n values that do not increase by 1; they increase by 6. Also, the n values begin at 0, not 1…. When this happens, the equation must change…. Exponential growth examples

The old equation is for patterns where n starts at 1, and increases by 1 For most application problems (ie word problems), we’ll use this new equation: Where: x measures time since the start y is the amount at time x, r is the common ratio, A 0 is the original amount (ie, the amount at time x = 0), and period is the amount by which the x values increase. (Note, sometimes x is replaced by t to emphasize that it measures time.) Exponential growth examples

So we get the function For this question, we can see that: r = 2 (doubles) A 0 = 1 period = 6 Exponential growth examples Number of hours since the start061218 Number of bacteria present1248 Ex. A certain type of bacteria doubles every six hours. The experiment begins with 1 bacteria.

Now let x = 50 and solve for y. There would be 322 bacteria at time x = 50 hours. a) How many bacteria are present after 50 hours? Exponential growth examples

Ex. A certain type of bacteria doubles every six hours. The experiment begins with 1 bacteria. b) When will there be 128 bacteria present? Now let y = 128 and solve for x. At x = 42 hours there will be 128 bacteria. Exponential growth examples

Ex. A certain type of bacteria doubles every six hours. The experiment begins with 1 bacteria. c) When will there be 200 bacteria present? Now let y = 200 and solve for x. But 200 cannot be written as a power with base 2. We’ll need our stronger tool, logarithms, to solve this one. Exponential growth examples

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