Pre Algebra (Period 2 & 4)

Exponents 4-2

2⁶ represents 2⋅2⋅2⋅2⋅2⋅2 = 64

the 2 in 2⁶ is the base
the 6 in 2⁶ is the exponent
and together 2⁶ it is the power!!
The base is used as a factor 6 times to produce 64

There are 3 forms here:
2⁶ is in exponential notation
2⋅2⋅2⋅2⋅2⋅2 is in expanded notation
64 is in standard notation

You must use integer rules-- even when raising a negative number to a power!!

odd power = negative
even power = positive
(-5)³ = (-5)(-5)(-5) = -125

(-5)⁴ = (-5)(-5)(-5)(-5) = 625


an odd number of negative signs or an odd power---> it's negative
an even number of negative signs or an even power ---> it's positive

(-2)³ = ( -2)(-2)(-2) = -8
(-2)⁴ = (-2)(-2)(-2)(-2) = 16

If there is a negative BUT NO parenthesis it is ALWAYS negative

-2⁵
is read as the opposite of 2⁵

Look at what the exponent is touching.. and in this case it is just the 2!!
You can also think of -2⁵ as (-1)-2⁵

so this is -1⋅2⋅2⋅2⋅2⋅2 = -32

-2⁴ = -2⋅2⋅2⋅2 or -1⋅2⋅2⋅2⋅2 = -16
BUT REMEMBER
(-2)⁴ = (-2)(-2)(-2)(-2) = +16


THis works with variables as well. WHen you substitute in for variables put the number in parenthesis!! Hugs are important in life-- and equally important in math!!
For example Solve for x
x⁴ - 10 when x = -2
(-2)⁴ - 10
+16 - 10 = 6
But what happens when the expression is
-x⁴ - 10? Still solve for x , when x = -2
This time
the exponent is touching just the x
-(-2)⁴ - 10
You must use PEMDAS and do the exponent first!!
(-2)⁴ = 16 so substitute that back in
-(16) - 10
becomes
-16 -10 = -26

What about 4(2y -3)² when y = 5
substitute in
4[2(5) -3]²
4(10-3)²
4(7)²
4(49) =196


How about
-2x³ + 4y . When x = -2 and y = 3

-2(-2)³ + 4(3)
-2(-8) + 12
16 + 12 = 28