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Basic Math concepts for a child



Exponential Notation
Exponents, or powers, are an important part of math as they are necessary to indicate that a number is multiplied by itself for a given number of times. 
This section will help you understand the notation used to indicate powers, or exponents. 
Important Things to Remember
  • Exponents are a "short cut" method of showing a number is multiplied by itself. 
  • Exponents can be shown to different ways.  Example: 
    •         x2 or x^2
  • Know the difference between -x^y and (-x)^y.  Example: 
    •         -34 = -(3)(3)(3)(3) = -81
        (-3)4 = (-3)(-3)(-3)(-3) = 81

The Tutorial
When dealing with exponents, remember that exponents are a "short cut" to show that a number is to be multiplied by itself a given number of times.  For example, x2 is the same as x * x.  The number or symbol (variable) that is to be multiplied by itself is called the base (in the example given above, the base is x), and the number or symbol showing how many times it is to be multiplied by itself is called the exponent or power (in the example above, the power is 2). 
Examples 
1.   Simplify:   (-5)^3


     Solution:  
     (-5)(-5)(-5)
   
     -125
  
Take note of the parentheses.
Realize that the problem is (-5) cubed.
The power (3) shows that the base (-5)
needs to multiplied by itself 3 times. 
2.   Simplify: -33 - (-3)2 + (-2)2


     Solution:  
     -(3)(3)(3) - (-3)(-3) + (-2)(-2)
     -27 - 9 + 4
     -32
  
Watch out for the first term,
which does not have parentheses around it.
Simplify each expression and then add the
terms for the final answer.

Evaluation of Exponentials
The evaluation of expressions containing exponents is very straightforward.  It is the same as the evaluation of any other expression.  The only thing to look out for is a negative number.
This section will help you understand how to evaluate expressions with exponents. 

  • Be sure to note parentheses when dealing with exponents.  Always evaluate anything inside parentheses first.  Example: 
    •         -34 = -(3)(3)(3)(3) = -81
        (-3)4 = (-3)(-3)(-3)(-3) = 81

The Tutorial
Evaluating expressions is something very common in algebra and is useful later on when you have to check solutions of equations.  Evaluating expressions with exponents is just as easy.  Just be sure to notice negative numbers and negative signs when dealing with exponents because they can make a big difference.
Example 
1.   Evaluate:   yx2z3
                 y = 3, x = 4, z = 2


     Solution:  
     (3)(4)2(2)3
     (3)(16)(8)
     384
  
Plug the numbers into the expression.
Simplify.


Theorems for Exponents
There are three different theorems that deal with exponents.
This section will help you better understand these theorems.

  • x0 = 1 when x does not equal 0.  x can be anything (except zero), including numbers, variables, or an equation.
  • x1 = x
  • x-n = 1/xn when x does not equal 0.  Example:
    •        x-2 = 1 / x2
  • Answers are considered simplified when exponents are positive unless otherwise noted. 

The Tutorial
There are three theorems that are special to exponents.  They are outlined below. 

Product Theorem for Exponents
If m and n are real numbers and x does not equal 0, xm * xn = xm+n

Quotient Theorem for Exponents
If m and n are real numbers and x does not equal 0, xm/xn = xm-n = 1/xn-m

Power Theorem for Exponents
If m and n are real numbers and x does not equal 0, (xm)n = xmn
Examples 
1.   Simplify:   x2y2x5y3


     Solution:  
     x2x5y2y3
     
     x7y5
   Rearrange the factors so
they are easier to deal with.
Use the Product Theorem to
simplify the expression. 
2.   Simplify:   x4
                 --
                 x6


     Solution:  
      1
     ----
     x6-4
     
      1
     ---
     x^2
  
Use the Quotient Theorem to
combine the numerator and denominator
into one term in the denominator. 
3.   Simplify:   (x-4)-2



     Solution:  
     x8
  
Use the Power Theorem to
multiply the two exponents into one.


Combining Like Terms Involving Exponents
Exponents add a new aspect to the operation of combining like terms, and thus make it slightly more complicated.
This section will help you understand how to combine like terms when they include exponents.

  • Rearranging the order of factors does not change their value.  Example: 
    •         x2yz5 = z5x2y
  • Power Theorem for Exponents, which states (xm)n = xmn.

The Tutorial
When you come across an expression with many terms, it is easier to deal with that expression when it has been simplified by adding like terms.  When doing this with an expression that contains exponents, the variables and their exponents have to be the same.
Example 
1.   Simplify:   x2yz5 + 2xy2z5 + 3z5x2y - 7y2xz5


     Solution:  
     x2yz5 + 3x2yz5 + 2xy2z5 - 7xy2z5
     
     4x2yz5 - 5xy2z5
   Rearrange the factors so
they are more easily identifiable as like terms.
Combine like terms and get this answer.


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