Download Everything I Know About Exponents

Everything I Know About Exponents
By: Alyson Vance
1) Represent repeated multiplication with exponents.
44 x 43 = (4 x 4 x 4 x 4) x (4 x 4 x 4)
2) Describe how powers represent repeated multiplication.
44 represents repeated multiplication by saying that 4 is repeated 4 times. So 44 is equal to
4x4x4x4.
3) Demonstrate the difference between the exponent and the base by building models of a
given power, such as 23 and 32.
The base in the equation is the number that you multiply. For example; in 23 the base is 2
because that is the number you are multiplying. The exponent in 23 is 3 because that is how
many times you are multiplying the base. 2x2x2 You take the base (2) and you multiply it as
many times as the exponent says (3).
Here is an example of 23:
23 is an example of an area model.
Here is an example of 32:
32 is an example of a volume model.
4) Demonstrate the difference between two given powers in which the exponent and the
base are interchanged by using repeated multiplication, such as 23 and 32.
If the numbers in each equation are the same but the bases and exponents are switched, (such
as 45 and 54) they will not result in the same answers. You can often get confused and think
that they are the same thing. But when you put it in repeated multiplication form, it's often
easier to tell. For example: 45= 4x4x4x4x4= 1,024
54= 5x5x5x5= 625
They result in different answers because the exponents are different and the bases are not the
same.
5) Evaluate powers with integral bases (excluding base 0) and whole number exponents.
-(5)4= (-5)x(-5)x(-5)x(-5)= 625
35= 3x3x3x3x3= 243
(-54)= (-1)x5x5x5x5= -625
6) Explain the role of parentheses in powers by evaluating a given set of powers such as
(-2)4, (-24) and -24
Brackets can change a lot in your equations. To show how they effect the equation, let's write it
as repeated multiplication. We will use these 3 numbers, (-3)6,(-36), and -36.
(-3)6= (-3)x(-3)x(-3)x(-3)x(-3)x(-3)= 729 By the exponent being on the outside the brackets, it
means that you multiply the base (-3) six times (exponent).
(-36)= (-1)x3x3x3x3x3x3= -729 By the exponent being on inside of the brackets, it means that
the negative sign becomes a coefficient of (-1) and that the three’s are positive.
-36= (-1)3x3x3x3x3x3= -729 By there being no brackets at all, the negative sign becomes a
coefficient of (-1) and the three’s become positive.
7) Explain the exponent laws for multiplying powers with the same base.
Product law:
When multiplying powers of the same base you:
1. Keep the base.
2. Add exponents together.
Example: 45 x 43= 4(5+3)= 48
8) Explain the exponent laws for dividing powers with the same base.
Quotient law:
When dividing powers of the same base you:
1. Keep the base.
2. Subtract the exponents.
Example: 45/43= 4(5-3)= 42
9) Explain the exponent laws for raising a product and quotient to an exponent.
Power law:
When taking a power and raising it to another exponent:
1. Keep the base.
2. Multiply exponents together.
Example: (45)3= 4(5x3)= 415
10) Explain the law for powers with an exponent of zero.
Zero exponent law:
When a power has an exponent of zero, the answer will always equal 1 (except when the base
is 0).
Example: 8000= 1
11) Use patterns to show that a power with an exponent of zero is equal to one.
55= 3,125
54= 625
53= 125
52= 25
5 1= 5
5 0= 1
When the exponent goes down by 1 you divide the answer by 5. So if 51 equals 5 then 50 will
equal 1 because you are taking the previous answer and dividing it by 5. So 5 divided by 5
equals 1.
12) I can apply the laws of exponents.
Product law:
23 x 2= 2(3+1)= 24= 16
54 x 53= 5(4+3)= 57= 78,125
Quotient law:
66 / 64 = 6(6-4)= 62= 36
47 / 45 = 4(7-5) = 42= 16
Power law:
(43 )6= 4(3x6)= 418= 68,719,476,736
(34)2= 3(4x2)= 38= 6,561
Zero exponent law:
70= 1 (When a power has an exponent of 0, it equals 1)
120= 1
13) I can identify the error in a simplification of an expression involving powers.
78 / 74 = 732
The simplification of this equation is wrong. This person used the power law and multiplied the
exponents where they should have used the quotient law and subtracted them. The correct
answer should have been: 78 / 74 = 7(8-4)= 74
14) Use the order of operations on expressions with powers.
The order of operation on expressions with powers is BEDMAS.
Examples:
35 + (4+5)2
=35 + 92
=125 + 81
=206
213 / [9-(2)3]
=213 / [9-(8)]
=213 / 1
=9,261 / 1
=9,261
1. Do what is in the brackets.
2. Do the exponents
3. Add numbers together.
1. Do the exponent inside brackets.
2. Do the rest of what is in the brackets.
3. Do the exponent on the right.
4. Divide.
15) Determine the sum and difference of two powers.
82 - 45= 64 - 1,024= -960
82 + 45= 64 + 1,024= 1,088
16) Identify the error in applying the order of operations in an incorrect solution.
(3x4)2 + 4 - 72
=122 + 4 - 72
=162- 72
= 92
=81
This persons mistake was that they forgot to do the exponents before the addition and
subtraction. They should have done it like this:
(3x4)2 + 4 - 72
=122 + 4 – 72
=144 + 4 – 49
=148 – 49
=99
17) Use powers to solve problems (measurement problems)I
f I have a cube with a side length of 4 cm, what is the surface area of the cube? What is the
volume of the cube?
SA of cube= 6(a2)
SA of cube with the side length of 4cm= 6(42)
SA= 6(16)
SA= 96
V of cube= a3
V of cube with the side length of 4 cm= 43
V= 64cm3
18) Use powers to solve problems (growth problems)
There are 2 single bacterium’s. Each bacterium double in number every hour. How many are
present after:
a. 1 hour?
2(41)
=2x4 =8
b. 2 hours?
2(48)
=2x65,536 =131,072
c. 3 hours?
2(412)
=2x16,777,216 = 33,554,432