10/6 3A Exponents - Desert Academyusers.desertacademy.org/balei/Math/1718/SL/SL1.AlgFuncCh3Exponent… · 10/6 3A Exponents One of the ... Simplifying Writing an expression in an

SL1.AlgFuncCh3Exponents.1718.notebook

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October 14, 2017

10/6 3A Exponents

One of the biggest issues with powers is being clear what the base is.

Evaluate

Thinking of this as 1 times 3, five times or (1)(3)(3)(3)(3)(3)may help you to understand negative exponents (later).

The exponent operates on the symbol that immediately precedes it!

3A: #1,3,6,7 (Exponential notation) 3B: #111 last 4, 12 (Exponent laws)

Zero Exponent Property: a0 = 1 for all a ≠ 0

Can your understanding of exponents help you evaluate 2–4 ?

means multiply 1 by a, n times

means divide 1 by a, n times

+1

+11

1

1

1

1

1

1

1

1

Objectives1. Review and understand exponent notation

We had a quiz on 10/5 after which we briefly covered sections 3A & 3B . Go through the HW well on 10/9

Powers of 10

Some notation to understand:

is the line through points A and B

is the ray starting at point A and extending (infinitely) through B

is the line segment connecting points A and B

is the length of the line segment between A and B

Do you know how to find the distance between two points?

/12

https://www.youtube.com/watch?v=bhofN1xX6u0


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October 14, 2017

10/6 3B Laws of Exponents

Zero Exponent Property: a0 = 1 for all a ≠ 0

Using your understanding of exponents, write down an equivalent power for each product.

Using your understanding of exponents, write down an equivalent power for each power.


Using your understanding of exponents, write down an equivalent power for each quotient.


Objectives1. Review and understand the origins of various exponent laws and properties


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October 14, 2017

10/6 3B (cont) Summary & Practice

For now, we will restrict ourselves to integer exponents. More on that next time...

Properties of Exponents

Let a and b be real numbers and m and n be integers. Then:

There are a lot of useful properties. Do not memorize them! Understand them!

This takes practice...

3A: #1,3,6,7 (Exponential notation) 3B: #111 last 4, 12 (Exponent laws)


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October 14, 2017

10/9 3C Rational Exponents

We have discussed integer exponents. Today we'll look at rational exponents (ie. fractional exponents). Consider, for example, the exponent of one half. Recall that we can think of exponents as the number of times that you multiply 1 by a base, say 3. 35 means multiply one by 3, five times. So what would it mean, for example, to multiply one by 3, a half a time?

Can we support this using properties we know?

The product of powers property can help. It implies that

But the number that multiplies by itself to get a already has a name:

So... and in general which is the nth root of a.

The properties that we have developed imply some other interesting things:

What happens if we cube the square root of 16?

What happens if we take the square root of 16 cubed?

Let's look at that using exponents:

means the (square root of 16) cubed...or the square root of (16 cubed)

In general, Rational Exponents

Some practice: Write as single powers of 2 or 3

Try a couple on your calculator:

3C: #15 last 4 (Rational exponents)3D.1: #12 last col (Algebraic expansion)Objectives

1. Understand and apply rational exponents

3A: #1,3,6,7 (Exponential notation) Present #6,73B: #[1,2,5,811] last 3, 12 (Exponent laws) Practice previous page. Questions?

Note change from green HW plan handout

We will continue at a fast pace. Hang in there.The quiz on this chapter will be next Thursday 10/19It will include a "speed quiz" of powers of 2 (up to 210) and perfect cubes (up to 103). Begin to memorize them now!


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October 14, 2017

10/9 3D.1 Expansion

Also know how to deal with different situations:

Practice is critical! Work thoroughly, make no move without absolutely knowing why.

Pay attention to the pattern (not the answers). Because next time we are going in the other direction (aka factorisation).

You try some:

Careful!!!

Guess what you can do this with exponents but you need to understand conceptually what factors are and how exponents work. Some examples:

Objectives1. Expand expressions involving integer and rational exponents

3C: #15 last 4 (Rational exponents)3D.1: #12 last col (Algebraic expansion)

. 1

HW is not a lot of practice. Do more problems until you are getting them correct the first time - 1 minute per problem maximum!

Recall the common properties of algebraic expansion.


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October 14, 2017

Rational Exponent Quizzes

Simplify the expressions.

Warm up: Simplify the expressions.


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October 14, 2017

10/10 3D.2 Factoring

Last time we expanded. This time, we will do the reverse of that ... factor. Let's review:

HW is not a lot of practice. Do more problems until you are getting them correct the first time 1 minute per problem maximum!

Question: Why? would it be useful to know how to factor these things?

Objectives1. Factor expressions involving

integer and rational exponents

3D.2: #15 last row, 6 (Factorization)3E: #15 last row (Exponential equations)QB: #1,2 (QB: Exp Equations)

. 2

A Key Idea: a2x = (ax)2 which we can treat the same way we would treat blah2

A huge skill, as important as dividing (it actually is dividing)

Factoring means writing an expression as a product of factors.

In essence we are undoing multiplying. Look for common factors in each term.

6x + 18 = 6(x + 3)

...but not just numbers! Common factors can be variables!You may remember some like this:

The same ideas hold with variables in the exponent:

You may be able to factor an expression even if it has no common factors!There are several methods to try:

1) Reverse FOILGuess and checkThe British method (for quadratics with a not equal to one)

2) Recognizing a special pattern3) Factor by grouping

Special Factoring Patterns

In SL, these ideas get extended in many directions, including with exponents:

Simplifying Writing an expression in an equivalent form We also need to know how to simplify expressions involving exponents:

3C: #15 last 4 (Rational exponents) Verbal quiz on all3D.1: #12 last col (Algebraic expansion) Present 2i,3fil

The quiz on this chapter will be next Thursday 10/19It will include a "speed quiz" of powers of 2 (up to 210) and perfect cubes (up to 103). Begin to memorize them now!

Return Function Quizzes


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October 14, 2017

10/10 3E Exponential Equations

Exponential expressions often occur in equations that we need/want to solve.

We've seen this a little bit before. In general, we may need to use logs. But we'll save that for the next chapter. Meanwhile, notice that there are some forms of exponential equations that we can solve without using logs. Suppose, for example, that we wanted to solve the equation 8 = 2n

Because this can be written as 23 = 2n we can mathematically support what we know to be true.

Equating Exponents

If you can rewrite an exponential equation to have powers of the same base on either side, you can equate the exponents to solve the equation.

Try a few

A few more. Notice how you can manipulate things. You're looking for:> One term on each side> The same base on each side (rewrite the bases)> Expressions in the exponents that you can set equal to each other> Also look for quadratic forms> There may not be solutions

Another idea: Notice the quadratic form. Factor and then find each zero.

This is the minimum homework. Do enough to become proficient.

Objectives1. Solve exponential equations by equating exponents

3D.2: #15 last row, 6 (Factorization)3E: #15 last row (Exponential equations)QB: #1,2 (QB: Exp Equations)

The quiz on this chapter will be next Thursday 10/19It will include a "speed quiz" of powers of 2 (up to 210) and perfect cubes (up to 103). Memorize them now!


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October 14, 2017

10/12 3F Graphs of Exponentials

Graphs help us to visualize relationships (functions). What does an exponential look like? Start with a table of values:

x

0 1

1 2

2 4

1 1/2

2 1/4

Notice that the function can never reach zero. It has an asymptote at y=0.What is the range of the function? y | y > 0What is the affect of changing the base?How about the parameter a?

The general exponential has four parameters that control its shape:

Understanding this, we can graph exponential functions.

To graph:> Plot the yintercept> Plot two other points (x = 1 and x = 1 are usually easy)> Know the general shape and connect

GGB Demo

3F: #1bd,26all (Exponential functions)Objectives1. Graph and understand graphs of

exponential functions.

The quiz on this chapter will be next Thursday 10/19It will include a "speed quiz" of powers of 2 (up to 210) and perfect cubes (up to 103). Know them forward and in reverse!

To find the function from a graph is not so easy because there are so many possibilities.

3F: #1bd,26all (Exponential functions)

Quick Quiz

1. Write as a single power of an appropriate base. [2 marks each]

2. Expand and simplify fully. [4 marks each]

A2 3(32a) A2

32x + 7(3x) + 10 A1A1A19x + 7(3x) + 10 A1

72x – 2 + 7–2x A1A1A149x – 2 + 1/49x A1

3D.2: #15 last row, 6 (Factorization) Present 1f,2hi,3ef,4gh,5i,6ab3E: #15 last row (Exponential equations) Take questionsQB: #1,2 (QB: Exp Equations) Present QB


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October 14, 2017

10/17 3G Exponential Growth

Imagine that 100 rabbits are dropped onto an island and they immediately start reproducing at the rate of 50% per year. Can you write a function that gives the population of rabbits at some number of years n after the initial drop off? Can you graph the function?

The population after the nth year is given by the function P(n) = 100(1.5)n

Let's look at the graph of this function:

Wow This is the equivalent of 3 child families (assuming no deaths, of course!) But you get the picture. More realistic: World population growth rate is approximately 1.17% per year:

Consider this context:

Here's the same data in another view. What does it communicate?

So ask yourself why some understanding of math is important....

Increasing a number by p% is equivalent to multiplying the number by 1 + p/100

Year Calculation Population at year's end

0 100 100

1 100 (1.5) 150

2 100(1.5)(1.5) 225

3 100(1.5)(1.5)(1.5) 337.5

... ... ...

n 100(1.5)(1.5)...(1.5) 100(1.5)n

Objectives1. Understand and use principles of

exponential growth and decay

3G.1: #2,4 (Exponential growth)3G.2: #1,3,4,5 (Exponential decay)3H: #1,3,6,7,1013 (The number e)Review 3C: #111 as needed

3F: #1bd,26all (Exponential functions) Present 4 (what did you learn?), 6abcd

Most of you would be wise to do some more work on sections 3C3E. Your quick quizzes showed basic understanding but careless mistakes under pressure. Do a lot of problems as fast as you can.

Quiz next time. Expect refresher questions (but don't sweat them)It will include a "speed quiz" of powers of 2 (up to 210) and perfect cubes (up to 103). Know them forward and in reverse!


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October 14, 2017

10/17 3G (cont) Exponential Decay

Exponential functions also describe decay. A common example is radioactive decay where, for example, a radioactive substance deteriorates (decays) at a rate of 5% per year. Now we have a "growth multiplier" (1 + r) that is less than 1. It's really (1 r).

The amount of the substance after n years is given by Wn = W0(0.95)n

Suppose we started with 50 grams. We can ask questions such as:

• How much is present after 10 years?• When will the amount left be equal to than 10 grams? (we'll leave

that for later!)

The exponent is not always an integer number like n. Let's look at some examples:

Radioactive materials are often described by their halflife. This is the amount of time it takes to reach half the initial weight. Given that some material has a half life of 1000 years, what is the annual rate of decay?

V = P(1 – 0.07)t

3000 = P(0.93)3

3000/(0.93)3 = $3,729.69

Thus the annual decay rate is 10.9993...=0.00069291 = .0693%

Exponential Growth

The amount after t years of something growing at a rate of r % per year is given by:

A(t) = A0 (1 + r)t where A0 is the initial amount present. r is called the growth rate and is written as a decimal. The value 1 + r is called the growth factor. This idea can be extended for any time period as long as t and r are given in the same units of time (years, months, seconds, etc.)


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October 14, 2017

10/17 3H The number e

Let's go back to compound interest again. You will recall that the formula for the amount of money in the bank after t years if you invest P dollars at an annual interest rate of r, compounded annually is given by:

A = P(1 + r)t compounded annually

If you compound, say, quarterly, you will be paid 4 times as often but the interest rate you receive each quarter is only a fourth of the annual interest rate. Thus, the formula becomes:

If you compound monthly, there are 12 periods. So the formula becomes:

A question arises: what if you compounded "continuously" meaning that we let the number of periods per year get larger and larger on to infinity! Let's look and see if there's a pattern. To do this, we need to generalize our equation first. Let's use n to represent the number of compoundings. Then we have:

To explore this, let's make a substitution . Can you show algebraically that

This is nice because we can explore the factor in square brackets. Notice that as n gets larger and larger, so does a (it's just n divided by a constant). So let's see what happens as a gets large.

a

1 2

10 2.593742

100 2.704814

1000 2.716924

10000 2.718146

100000 2.718268

1000000 2.71828

10000000 2.718282

100000000 2.718282

This number is the natural number called "e" after Leonhard Euler who discovered it. It shows up in nature in many strange and interesting ways.

It also is the sum of

It is irrational!

So...when we have continuous compounding (or continuous growth) a gets very large and our equation:

becomesA(t) = Pert

There will be a quiz on Exponents, powers of 2, and perfcect cubes next time

Continuous Exponential Growth

The amount after t years of something growing continuously at a rate of r % per year is given by:

A(t) = Pert where P is the initial amount present. r is the growth rate (r > 1) or the decay rate (r < 1) and is the decimal % change in one unit of time (t).

Question: Suppose you have 10 grams of Somestuffium that grows continuously at a rate of 50% per year. How much will you have after one year? Your friend has 10 grams of Otherjunkium that decays continuously at a rate of 50% per year. How much will your friend have after one year?

You: 10e(0.5)(1) = 10e0.5 = ~16.5 gramsYour friend: 10e(–0.5)(1) = 10e–0.5 = ~6.07 grams

3G.1: #2,4 (Exponential growth)3G.2: #1,3,4,5 (Exponential decay)3H: #1,3,6,7b,1013 (The number e)Review 3C: #111 as needed

Spread this HW out!

Objectives1. Understand the origins of the number e.2. Use e in continuous growth and decay problems.


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October 14, 2017

Factoring Review


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October 14, 2017

Integer Exponent Practice

Some practice to warm up...simplify as many of the highlighted expressions as you can


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October 14, 2017

Short Exponent Quiz

1. Match each equation to its corresponding graph

2. Solve the equation 33x – 7 = 8112 – 3x

3. Michele invested 1500 francs at an annual rate of interest of 5.25 percent,compounded annually.(a) Find the value of Michele’s investment after 3 years. Give your answer to thenearest franc.(b) How many complete years will it take for Michele’s initial investment to double invalue?(c) What should the interest rate be if Michele’s initial investment were to double invalue in 10 years?

Geogebra Exponentials

Exponent Quiz

a=C A1b=E A1c=A A1d=B A1e=D A1

x = 55/15 = 11/3 (M1)A1

r = 7.18% (M1)A1

12 years @6.25%; 14 years @5.25% (M1)A1

1439 for [email protected]%; 1749 for [email protected]% (M1)A1

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10/6 3A Exponents - Desert Academyusers.desertacademy.org/balei/Math/1718/SL/SL1.AlgFuncCh3Exponent… · 10/6 3A Exponents One of the ... Simplifying Writing an expression in an