Notes

The first step in simplifying expressions is combining like terms. Like terms are terms that have identical variable parts.
Combining like terms is the process of adding or subtracting like terms to simplify an expression.
In order to add or subtract, you must have like terms. This is not the case for multiplication or division. Any two terms can be multiplied or divided; they do not have to be like terms.
When adding or subtracting expressions, you may need to find the factors of each term before simplifying. Factors that are common to two or more numbers or values are referred to as common factors.
Example 1: Find a common factor for the expressions 9(2x + y) and 4(2x + y).
Factors of 9(2x + y): 9 and 2x + y
Factors of 4(2x + y): 4 and 2x + y
The common factor is 2x + y.
Example 2: Find a common factor for the expressions xz and 3z.
Factors of xz: x and z
Factors of 3z: 3 and z
The common factor is z.
Example 3: Find the factors of 3yz.
3 and yz are factors, since 3 • yz = 3yz.
3y and z are factors, since 3y • z = 3yz.
3z and y are factors, since 3z • y = 3zy, or 3yz.
So, the factors of 3yz are 3, 3y, 3z, y, z, and yz.
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When Next is selected, the following is shown: Step 1: Group the like terms together. The first line of equation is shown as 2 x plus 5 y plus 2 z plus 2 y plus z plus 3 x. Then, 2 x and plus 3 x are copied down to the second line as equals 2 x plus 3 x. Plus 5 y and plus 2 y are copied down to the second line next to the previous 2 terms as plus 2 y plus 5 y. Plus 2 z and plus z are copied down to the second line next to the previous terms as plus z plus 2 z. So, the second line becomes equals 2 x plus 3 x plus 2 y plus 5 y plus z plus 2 z.
When Next is selected, the following is shown: Step 2: Combine the like terms by adding the numbers in front of the variables. The previous line of equation is copied down. 2 x plus 3 x from the previous line is copied down to the next line and morph into 5 x. Plus 2 y plus 5 y from the previous line is copied down to the next line and morph into 7 y. Plus z plus 2 z from the previous line is copied down to the next line and morph into 3 z. So, the last line becomes equals 5 x plus 7 y plus 3 z.
When Next is selected, the following is shown: Solution: 2 x plus 5 y plus 2 z plus 2 y plus z plus 3 x equals 5 x plus 7 y plus 3 z.
The expression a to the third power times a to the fifth power is given. The power indicates the number of times each a is multiplied by itself. So, a to the third power becomes a times a times a. Then, a to the fifth power becomes a times a times a times a times a. This can be simplified by multiplying all of the a’s together. That gives us a times a times a times a times a times a times a times a. If the number of a’s being multiplied is counted, it will be clear that there are eight a’s being multiplied together. The number eight can be represented as the exponent giving us a to the eighth power. Go back to the original expression of a to the third power times a to the fifth power. Notice that since both bases are the same, the exponents can simply be added. 3 plus 5 equals 8.
Here is another example. Take 5 x y times x to the third power times y to the fifth power. Identify how many of each variable is resent by placing exponents by each base. This results in 5 times x to the first power, times y to the first power, times x to the third power, times y to the fifth power. The like bases can have their exponents added, so this yields 5 times x to the 1 plus 3 power, times y to the 1 plus 5 power. Simplifying the exponents results in 5 times x to the fourth power, times y to the sixth power.
Notice three things. The expression can be expanded to find its value, or the exponents can be added together. All variables have an exponent. If no exponent is visible, then it has a value of 1.When Next is selected, the following is shown: Find 2 x minus y minus 3 z if x equals 4, y equals negative 3, and z equals negative 1. Here again you will start by substituting in for x, y and z.
When Next is selected, the following is shown: Expression 1 is 2 x minus y minus 3 z. Then, expression 1 is copied down as expression 2 right below it. All the variables x, y, and z are faded off and replaced with parentheses that are blank inside. Then, the blank that was x is filled in with 4. The blank that was y is filled in with negative 3. The blank that was z is filled in with negative 1.
When Next is selected, the following is shown: Remember to do the multiplication before the addition or subtraction and be careful with the signs.
When Next is selected, the following is shown: Expression 2 is copied down as expression 3. The term minus negative 3 starts flashing. A callout that points to it says the following: Look at this as negative 1 times negative 3 because of the understood 1. Minus negative 3 is plus 3.
When Next is selected, the following is shown: The previous term stops flashing. The term minus 3 times negative 1 starts flashing. A callout that points to it says the following: Look at this as negative 3 times negative 1 because the subtraction sign is also the negative for the 3.
When Next is selected, the following is shown: Expression 3 is copied down as expression 4. The term 2 times 4 is morphed into 8. The term minus negative 3 is morphed into plus 3. The term minus 3 times negative 1 is morphed into plus 3. So, expression 4 becomes 8 plus 3 plus 3, which is 14.When combining like terms, it is important to remember the following:
•Variable parts must be identical.
2x and 3x are like terms.
2x and 3y are not like terms.
•2x plus 3x is the same as saying “two x’s plus three more x’s.”
2x + 3x = 5x
When using exponential expressions, it is important to remember the following:
•Coefficients are the numbers that come before the variable telling you how many times the variable has been added.
3x
•Factors are values you can multiply together to get a product.
1, 2, 3, and 6 are factors of 6.
1, 2, 2x, x, x2, and 2x2 are factors of 2x2.
•The exponent is the small number to the right of the value and tells you how many times the value must be multiplied by itself.
72 = (7)(7)
y4 = (y)(y)(y)(y)
•If you do not see a variable or an exponent with a base number, the variable and the exponent are both understood to be 1.
3xy = 3(x)1(y)1
You can substitute values in for the variable, and then follow the order of operations to simplify the expression.
Find 2xy + 3y if x = 6 and y = –2.
2(6)(–2) + 3(–2)
= –24 + (–6)
= –24 – 6
= –30