Pre Algebra
Section 3.4
Unit 3 - Decimals
Section 4 – Exponents, Order of Operations , Square Roots
Decimals with Exponents
Exponents work the same way with decimals as they do with Integers and Fractions.
Example 1)
. 2.3
= 2.3 ∙ 2.3
= 5.29
Example 2)
−. Notice everything in the parentheses repeats,
including the negative.
−3.4
= −3.4−3.4
= 11.56
Example 3)
−. −1.5
= −1.5 ∙ 1.5
Notice only the number itself repeats, not the negative.
= −2.25
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Pre Algebra
Section 3.4
Example 4)
−. −1.4
= −1.4−1.4−1.4
Notice everything in the parentheses repeats 3 times.
= 1.96−1.4
Multiply the first two numbers.
= −2.744
Multiply the remaining numbers.
Example 5)
−−. . −−1.2 3.3
= −−1.2−1.23.33.33.3
Following PEMDAS , we evaluate both exponents first.
= −1.4435.937
Notice the negative in front had no exponent applied.
= −51.74928
Order of Operations
Order of operations are applied the same way to decimals as they were to Integers and fractions. We
can use PEMDAS to remind us of the order.
Recall from 1.4
The order that we are expected to follow in called The order of operations
1.
2.
3.
4.
Start with grouping symbols (parenthesis, etc)
Simplify any exponents
Complete any multiplication and Division – going from left to right.
Complete and addition and subtraction – going from left to right.
Example 6)
. + . . 0.25 + 0.134.3
= 0.25 + 0.559
We multiply before we add.
= 0.809
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Pre Algebra
Section 3.4
Example 7)
. ÷ . ∙ . We divide before we multiply because
Multiplication and division have the
same priority and we work left to right.
2.25 ÷ 0.5 ∙ 6.23
= 4.5 ∙ 6.23
= 28.035
Example 8)
. − ∙ . 22.03 − 3 ∙ 0.5
= 2 ∙ 4.1209 − 3 ∙ 0.5
First we compute the exponent.
= 8.2418 − 1.5
Then both multiplications.
= 6.7418
Last we subtract.
Finding the Area of a circle
To Find the Area of a circle we use the formula
= !
"ℎ$!$
For this class we will use the approximation
= 3.14159265 … &'( ! = !&()*+
≈ 3.14.
Example 9)
Find the area of the following circle
2m
In this example we are given the radius so we
will use the formula = ! =
!
=
2-
≈ 3.14 ∙ 2- ≈ 3.14 ∙ 4- Using PEMDAS
≈ 12.56- American River College
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Pre Algebra
Section 3.4
Example 10)
Find the area of a circle whose diameter is 5 in.
We will still use =
.
! , how ever we have to find r first. Since = , / = 01 = . 01.
!
=
≈ 3.142.5 )'
2.5)' ∙ 2.5)' = 2.5 ∙ 2.5 ∙ )' ∙ )' = 6.25)'
≈ 3.14 ∙ 6.25)'
≈ 19.625)'
Calculating a Mean
One application of order of operations is calculating an average, called the mean. Say you took 3 tests
and scored 77, 81, and 84. If all the tests were out of 100, then to calculate your average you will add
the 3 test scores together and divide by the number of test scores.
22345346
We divide by 3 because we are averaging 3 scores
=
6
= 80.666666666….
7
= 80. 6
Or rounded to the nearest whole 81.
Example 11)
Find the average of 77,88, 78 , and 81.
22344324345
6
=
6
6
We divide by 4 because we are averaging 4 scores.
Following PEMDAS, first we add since the top is grouped together.
= 81
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Pre Algebra
Section 3.4
Square roots
Perfect squares are numbers that are the product of a number and itself. They are 1, 4, 9, 16, 25, 36,
49,…
1 = 1∙1
4 = 2∙2
9 = 3∙3
16 = 4 ∙ 4
25 = 5 ∙ 5
36 = 6 ∙ 6
49 = 7 ∙ 7
64 = 8 ∙ 8
81 = 9 ∙ 9
100 = 10 ∙ 10
121 = 11 ∙ 11
144 = 12 ∙ 12
169 = 13 ∙ 13
196 = 14 ∙ 14
225 = 15 ∙ 15
⋮
400 = 20 ∙ 20
⋮
900 = 30 ∙ 30
⋮
The square root of a number is the number that you would times itself by to get the number. For
perfect squares the numbers are nice and clean.
√9 = 3
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This is read “The square root of 9 is 3”. Note 9 = 3 ∙ 3
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Pre Algebra
Section 3.4
Example 12)
√
√100
=√10 ∙ 10
= 10
Example 13)
√
√25
=√5 ∙ 5
=5
If I were to ask : = 9, then there would be two answers. Notice 3 ∙ 3 = 9 and −3 ∙ −3 = 9. So there
are two square roots of 9. One is 3 and the other is -3. In the above we found the positive root. To
obtain the negative root we need to add a negative sign in front of the root.
Example 14 )
−√
−√36
Notice this example has the negative in front of the root.
= −√6 ∙ 6
=−6
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Therefore the answer is negative.
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Pre Algebra
Section 3.4
Exercise 3.4
NAME:___________________________________
Simplify the following
1. −2.1
2.−0.9
3.−1. 5
4. 1.53.2
5.−1.2 1.1
6. −2.1 0.2
7. 1.1 − 5.61.3
8.2.4 − 3.2 + 4.5
9.7.2 ÷ 0.5 ∙ 1.1
10.20.8 − 6.3 ÷ 0.3
11.4.1 − 3.6 + 4 ∙ .02
12.4.7 − 1.2 + 4 ÷ 0.2
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Pre Algebra
Section 3.4
For the following calculate the Area of each circle with the given dimension.
13. r = 3m
14. d=7in
15. d=6ft
For problems 13-15 compute the mean of the scores listed, round to the nearest hundredth if necessary.
16. 62,77,75,81
17. 88,81,85
18. 90,82,98
Find the following Square roots
19. √121
22. −√144
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20.√64
21.√225
23.−√196
140
24. −√1600
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