Using the Product Rule with Exponents

When you multiply constants (variables) that have the same base, you add the exponents... but keep the base unchanged.

For example:

x^2c · x^3 = x^(2+3) = x^5
(x · x) (x · x · x) = x^5

"X" squared times "X" cubed equals "X" to the fifth power.



Try a few more.

1)  p^5 · p^4 =
2)  2t^2 · 3t^4
3)  r^2 · 2^3 · r^5
4)  3x^2 · 2x^5 · x^4
5)  (p^2)(3p^4)(3p^2)


Answers:

1)  p^9
2)  6t^6
3)  2r^10
4)  6r^11
5)  9p^8

Simplify and Solve Using the Addition Principal of Equality

4 ( 8 - 15) + (-10) =  x - 7




Answer:
4 ( 8 - 15) + (-10) =  x - 7
32 - 60 + (-10) = x - 7
-28 + (-10) = x - 7
-38 = x - 7
-38 + 7 = x -7 + 7
-31 = x + 0
-31 = x

Solving Equations Using the Addition Principle of Equality

Can you find the error in the following problem?

5² + (4 - 8) = x + 15

25 + 4 = x + 15

29 = x + 15

29 + (-15) =  x + 15 + (-15)

14 = x + 0

14 = x

Practice Solving Simple Equations Using the Addition Property of Equality

It is important to practice the addition property of equality.  See below and solve five simple equations using the addition property of equality.

Practice Problem #1
x - 11 = 41

Practice Problem #2
x - 17 = -35

Practice Problem #3
84 = 40 + x

Practice Problem #4
45 = -15 + x

Practice Problem #5
-21 = -52 + x


Answers:

Practice Problem #1
x - 11 = 41
x - 11 + 11 = 41 + 11
x + 0 = 52
x = 52

check

52 - 11 = 41
41 = 41

Practice Problem #2
x - 17 = -35
x - 17 + 17 = -35 + 17
x + 0 = -18
x = -18

check

-18 - 17 = -35
-35 = -35

Practice Problem #3
84 = 40 + x
84 + ( - 40) = 40 + (-40) + x
44 =  0 + x
44 = x

check
84 = 40 + 44
84 = 84

Practice Problem #4
45 = -15 + x
45 + 15 = -15 + 15 + x
60 = 0 + x
60 = x

check
45 = -15 + 60
45 =  45


Practice Problem #5
-21 = -52 + x
-21 + 52 = -52 + 52 + x
31 = 0 + x
31 = x

check
-21 = -52 + 31
-21 = -21

Solving Equations Using the Addition Property of Equality

The addition principle of equality states that if a = b, then a + c = b + c.

When you solve equations using this addition principle of equality, you need to use the additive inverse property.  In other words, you must add the same number to both sides of an equation. 

Example #1:

x - 5 = 10

x - 5 + 5 = 10 + 5   We add the opposite of (-5) to both sides of the equation.

x + 0 = 15                We simplify     -5 + 5 = 0.

x = 15                       The solution is x = 15

 

To check the answer, simply substitute 15 in for x, in the original equation and solve.

15 - 5 = 10

10 = 10

 

Example #2:

x + 12 = -5

x + 12 + (- 12) = -5 + (- 12)   We add the opposite of (+12) to both sides of the equation.

x + 0 = -17                    We simplify +12 - 12 = 0.

x = -17                           The solution is x = -17

Check our answer.

(-17) + 12 = -5

-5 = -5