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MA 1128: Lecture 02 – 5/27/2015

Exponents

Scientific Notation

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Exponents

Exponents are used to indicate how many copies of a number are to be multiplied together.

For example,

(-2)5

= (-2)(-2)(-2)(-2)(-2)

I like to deal with the signs separately.

In this example, five negatives is negative, so

(-2)5 = -25 = -32

Similarly, since four negatives is positive,

(-2)4 = 24 = 16

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Multiplying Exponents

Since the exponent indicates the number of copies to be multiplied,

3537

Means five copies of 3 and seven more copies of 3 are to multiplied,

for a total of 12.

Therefore, we can write

3537 = 35+7 = 312

In 35, the 3 is called the base, and the 5 is called the exponent.

Remember: When multiplying exponent expressions with the same base,

you can simply add the exponents.

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This comes in handy with variables.

As I said before, it’s hard to implement the order of operations when we have variables, since we don’t have specific numbers to compute with.

But they follow the same rules, since variables represent numbers.

It must be true, for example, that

x2x5 = x2+5 = x7

Of course, this rule does not help us with

3472 or x3y2

Since the bases are different,

Or with

25 + 23

Since we’re not multiplying.

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Negative Exponents

A negative exponent is our way of indicating how many copies of a number are to be divided.

For example,

434-2

Means multiply three 4’s and divide two 4’s.

Dividing by 4 is the same thing as multiplying by ¼, so

434-2 = 4 4 4 ¼ ¼When multiplying, order does not matter (multiplication is commutative)

= 4 4 ¼ 4 ¼ = 4 (4 ¼) (4 ¼) = 4 1 1 = 4

Note that this follows the adding-the-exponents rule

434-2 = 43 2 = 41 = 4

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In terms of cancellation.

We can look at this last example in terms of cancellation.

Negative exponents mean divide, multiplying by 4-2

is the same as dividing by 42 (with a positive exponent).

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41

4

44

444

44

444

When you cancel everything with multiplication or division, you’re left with 1.

2

323

4

444

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Exponents Raised to Exponents

If we have something like

(32)4

The exponent 4 means four copies of 32 should be multiplied.

= (32) (32) (32) (32)

Since 32 means two copies of 3,

= (33) (33) (33) (33)

Two copies four times means that we have eight copies all together.

Therefore,

(32)4 = 32 4 = 38

Remember: If we have an exponent raised to another exponent,

We can simplify by multiplying the exponents.

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This applies equally to negative exponents

The expression (32)4

Means divide by 32 four times.

All total, we should divide by 3 eight times.

(32) 4 = 3(2)( 4) = 3 8

Finally, we can make sense of zero exponent as follows.

Since

x2x2 = x2 2 = x0

And x2 divided by x2 must be 1,

We can conclude that

x0 = 1

If x is zero, this doesn’t make sense, but otherwise, we’ll always have x0 = 1.

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Practice Exponent Problems

Simplify each of the following expressions involving exponents.

1. 42 43

2. 33 35

3. 52 53

4. 70

5. (62)5

6. (23)2

Click for answers.

1) 45 = 1,024 (either answer is OK); 2) 38 = 6,561; 3) 51 = 5; 4) 1;

5) 610 = 60,466,176; 6) 26 = 64.

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More Examples

Suppose we are dividing by 52. That is, we’re dividing by a negative exponent.

On it’s own, 52 is 1 divided by 52.

We’re dividing by a fraction, and when we divide by fractions, we invert and multiply.

Therefore, dividing by 52 must be the same as multiplying by 52.

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62555555

5 422222

2

Remember: Within a single fraction with only multiplication in the numerator and denominator, you can move anything from the bottom to the top, or vice versa, just remember that the sign on the exponent changes.

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Scientific Notation

Scientific notation is used to express very large or very small numbers in a compact notation.

What I’ll refer to as correct scientific notation is a decimal number, greater than or equal to 1 and less than 10, times some power of 10.

For example, 2.17 102 is in correct scientific notation.

Since 102 = 100, we have that 2.17 102 = 217.

Most people look at the exponent on the 10, which is 2, and associate this with moving the decimal point two to the right.

Very small numbers will have a negative exponent on the 10.

5.261 109 = 0.000000005261

Here, we moved the decimal point nine to the left.

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Converting into Scientific Notation

When converting a number into scientific notation, we’ll move the decimal point the opposite way.

Whether the exponent on the 10 is positive or negative can be hard to remember,

but if you keep in mind whether the number is really big or really small, you should be OK.

For example, consider 2,056,000.

This is a really big number, and we want one non-zero digit to the left the decimal point.

Therefore, the decimal point has to move six places.

2,056,000 = 2.056 106,

and we know that the exponent is positive 6, because this is a big number.

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More Examples

Consider the number 0.000276

The decimal point must move four places to get one non-zero digit to the left of the decimal point, and this is a small number. Therefore,

0.000276 = 2.76 104

Now, let’s convert 65,020,000,000 into scientific notation.

This is a big number, so we’ll have a positive exponent.

We need to move the decimal point 10 places, so the exponent will be 10.

6.502 1010

When converting to or from scientific notation, the exponent agrees with the number of places the decimal point moved. Make sure that if you start with a big number, then you end with a big number.

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Multiplying and Dividing with Scientific Notation

Scientific notation is easy to work with. It wouldn’t be used otherwise.

Suppose we want to multiply two numbers in scientific notation. For example

(2.35 103)(1.22 107)

This really is just four numbers being multiplied together.

2.35 103 1.22 107

If we’re only multiplying, order doesn’t matter, so we can rewrite this as2.35 1.22 103 107 = 2.867 1010

We just multiplied the first numbers and multiplied the powers of 10, probably using a calculator.

The one thing we need to be careful of is that in correct scientific notation, there is exactly one non-zero digit to the left of the decimal point.

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More Examples

When multiplying, it’s possible for the two decimal numbers to multiply to something bigger than 10.

Consider this product

(6.32 102)(5.1 105)

Multiplying as we did in the previous example, we get

32.232 107

This is fine, except we now have two digits to the left of the decimal point.We need to move the decimal point to the left one place.

3.2232 10 107 = 3.2232 108

Remember that when you make the adjustment to correct scientific notation, you need to make sure the size of the number stays the same.

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More Examples

Here’s another example.

(4.76 108)(6.02 102)

The negative exponent is no problem. Just multiply the corresponding parts.

28.6552 106 = 2.86552 105

In the adjustment, we made the decimal part smaller by a factor of 10, the power of 10 must get bigger.

Here’s one more.

(4.01 104)(3.9 107) = 15.639 1011 = 1.5639 1012

Division works pretty much the same way, as you’ll see on the next slide

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Consider the following division problem.

As with multiplication, we divide the decimal parts and divide the powers of 10.

1.7857 105

Here we rounded to four decimal places.

Here’s another example.

Examples with Division

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3

8

105.3

1025.6

547

3

105352.41045352.0101.7

1022.3

Note that we adjusted the decimal part bigger, so the power of 10 had to get smaller.

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Practice Problems with Scientific Notation

Convert to or from correct scientific notation.

1. 2.77 108

2. 3.511 103

3. 7.923 105

4. 1.01 1013

5. 3,980,000

6. 89,010,000,000

7. 0.002133

8. 0.0000001013

Click for answers

1) 277,000,000; 2) 0.003511; 3) 0.00007923; 4) 10,100,000,000,000;

5) 3.98 106; 6) 8.901 1010; 7) 2.133 103; 8) 1.013 107.

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More Practice Problems with Scientific Notation

Multiply or divide and write your answer in correct scientific notation.

1. (3.13 102)(2.17 103)

2. (6.98 103)(2.5 105)

3. (3.92 105)(8.52 102)

4. (1.01 102)(2.65 103)

5. (5.13 102) (2.77 105) (I’m being lazy, and not using the division bar.)

6. (1.16 107)/(4.03 103) (Here’s another way to indicate division.)

Click for answers

All answers are rounded to two decimal places. Make sure you’re rounding correctly.

1) 6.79 105; 2) 1.75 103; 3) 3.34 102; 4) 2.68 105;

5) 1.85 103; 6) 2.88 103.

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