Simplification Rules for Algebra using Exponents

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Data Science Math Skills

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Duke University

Data Science Math Skills

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Data science courses contain math—no avoiding that! This course is designed to teach learners the basic math you will need in order to be successful in almost any data science math course and was created for learners who have basic math skills but may not have taken algebra or pre-calculus. Data Science Math Skills introduces the core math that data science is built upon, with no extra complexity, introducing unfamiliar ideas and math symbols one-at-a-time. Learners who complete this course will master the vocabulary, notation, concepts, and algebra rules that all data scientists must know before moving on to more advanced material. Topics include: ~Set theory, including Venn diagrams ~Properties of the real number line ~Interval notation and algebra with inequalities ~Uses for summation and Sigma notation ~Math on the Cartesian (x,y) plane, slope and distance formulas ~Graphing and describing functions and their inverses on the x-y plane, ~The concept of instantaneous rate of change and tangent lines to a curve ~Exponents, logarithms, and the natural log function. ~Probability theory, including Bayes’ theorem. While this course is intended as a general introduction to the math skills needed for data science, it can be considered a prerequisite for learners interested in the course, "Mastering Data Analysis in Excel," which is part of the Excel to MySQL Data Science Specialization. Learners who master Data Science Math Skills will be fully prepared for success with the more advanced math concepts introduced in "Mastering Data Analysis in Excel." Good luck and we hope you enjoy the course!

從本節課中

Measuring Rates of Change

This module begins a very gentle introduction to the calculus concept of the derivative. The first lesson, "This is About the Derivative Stuff," will give basic definitions, work a few examples, and show you how to apply these concepts to the real-world problem of optimization. We then turn to exponents and logarithms, and explain the rules and notation for these math tools. Finally we learn about the rate of change of continuous growth, and the special constant known as “e” that captures this concept in a single number—near 2.718.

Using Integer Exponents7:48

Simplification Rules for Algebra using Exponents11:08

How Logarithms and Exponents are Related12:36

The Change of Base Formula4:23

The Rate of Growth of Continuous Processes11:09

與講師見面

Daniel Egger
Executive in Residence and Director, Center for Quantitative Modeling
Pratt School of Engineering, Duke University
Paul Bendich
Assistant research professor of Mathematics; Associate Director for Curricular Engagement at the Information Initiative at Duke
Mathematics

There are five simplification rules for

keeping notation straight and doing algebra using exponents.

If you memorize and practice these five rules, you will be able to understand and

solve just about any algebra problems based on integer exponents.

We'll mention all five rules briefly.

Then go through a number of examples of simplifying exponential expressions,

discussing for each which of the five rules to apply.

Note that we use the word power in these rules in a special sense,

to mean the value of an exponent.

Our first rule is the multiplication rule.

When taking the product of the same factor

raised to different exponents, sum the exponents.

And this looks like this.

x to the n, and then the same factor again raised to a different exponent.

And this is = x raised

to the sum (n + m).

The second rule is the power to a power rule.

The idea here is that you have a number that contains an exponent.

And you're raising the entire thing to an exponent.

And the way that you simplify this is that you take the product of the exponents.

And that becomes your power.

The product to a power rule, Is relevant when we have two different factors.

And they are raised to a common exponent.

And what we do is distribute the exponent.

So we have each element of the product

raised separately to that exponent.

And let me illustrate that because that might not be completely obvious.

Let's say we have (2*3) to the 3rd.

Well, it should be clear that that would be equal to (2*3)(2*3)(2*3).

And then we gather up the 2s, so

we have 2 to the 3 times 3 to the 3, okay?

Now let's consider the fraction to a power.

Situation here is that we have one number on top, and another number on the bottom,

and we're raising the entire fraction to an exponent.

And again, we distribute the exponents.

So this becomes = x raised

to the n / y to the n.

When raising a ratio of two integers to a power,

distribute the exponent to each number.

Finally we have the division and negative powers rule, and this works as follows.

If we have x to the n / x to the m,

this is the same thing as x raised to the (n-m).

Now you might notice that this really combines rules we already have.

Because what we're really doing here is we're saying,

when we have x to the n x to the -m, that this is the same

as x to the (n-m), Okay?

Now we're going to work through some examples.

And you can pause the video.

And see if you can identify which rule to apply to simplify and solve the equation.

What is (7 to the 3rd)(7 to the 7th)?

Well, we apply the multiplication rule because we have a shared factor of 7.

So this is 7 to the (3+7), or 7 to the 10th.

What about (4 to the 3rd) raised to the 5th?

Well, here we have a power raised to a power.

So we have 4 to the 3 times 5,

or 4 to the 15th.

What about (8*9) to the 7th?

Well, here we distribute the exponent using the product to a power rule.

And we have 8 to the 7th times 9 to the 7th.

This is an example of when our scientific notation might be useful.

Because this is 1.00306x10 to the 13th.

What about (2/7) to the 3rd?

This is an opportunity to apply the fraction to a power rule.

We have 2 to the 3rd / 7 to the 3rd, or

0.023323615.

Now what about 10 to the 5th / 10 to the 3rd?

This is = 10 to the 5th times 10 to the -3,

which is = 10 to the 5-3.

Which is = 10 squared,

which is = 100.

Now let's try some slightly harder examples.

The way that I would tackle a problem like this is I would

isolate each separate factor.

So we have x to the 3rd / x to the 3rd,

we have y to the 4th / y to the 5th,

and we have z to the 5th / z squared.

Using our, Negative exponents rules, this is equal to x to the 3-3,

y to the 4-5 z to the 5-2.

Which is simply equal to x to the 0, which is equal to 1.

So we have 1 times y

to the -1 z cubed.

Or if you prefer, z cubed over y.

Let's try one more.

Here we'll start by isolating each factor, and then we'll do the -1 at the very end.

So we have the product to a power rule.

So we have x squared y squared on the top.

And x to the -3 y squared on the bottom.

So this is going to be = x to

the 2--3 times y to the 2-2,

so this is = x to the 5th.

However, We have multiplication by -1,

so this is = x to the -5, or 1 / x to the 5th.

And now you should try some on your own for on the practice quiz.

We want to touch briefly on one more topic.

Which is how you handle an exponent that is, itself, a fraction.

The answer is that you treat it as two separate operations.

Where the upper number is a standard exponent and the lower number is a root.

So in the example we have here, we have the 8 raised to the 2/3rds power.

What that means is 8 squared, cube root of that.

Or it's equally accurate to say the cubed root of 8 squared.

The order does not matter.

So let's see, what would that be?

Well, the cubed route of 8 =

2 because 2*2*2, or

2 to the 3rd = 8.

So we would have the cubed route of 8,

which is 2, squared, which would be equal to 4.

Would be equally accurate to say that we have 8 squared, which is = 64.

And we take the cubed root of that, and that would be =,

Cubed root of 4*4*4,

which would be = 4.

Let's consider another example, 125 to the 4/3,

We would take the cubed root of 125, which is = 5.

And we would raise that to the 4th power.

So that's 5*5*5*5, which is = 625.

So as long as you treat each part of a rational, or

fractioned, exponent as a separate operation,

you should be able to do these problems without too much trouble.

So that concludes our exponent rules.

And what I suggest is that you simply practice them a little bit.

And they will become second nature to you.

And they will not seem difficult if you just work some problems and practice.

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