Transcript 1-7 Simplifying Expressions
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1-7 Simplifying Expressions Warm Up Evaluate.
1. 4 2 3. –2 3 16 – 8 2. |5 – 16| 4. |3 – 7| 4 11
Translate each word phrase into a numerical or algebraic expression.
5. The product of 8 and 6 8 6 6. The difference of 10y and 4
Simplify each fraction.
10y – 4
7.
8
8.
1-7 Simplifying Expressions California Standards 1.0 Students use properties of numbers to demonstrate whether assertions are true or false. 25.1 Students use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions.
1-7 Simplifying Expressions
Vocabulary
order of operations terms like terms coefficient
1-7 Simplifying Expressions
When an expression contains more than one operation, the order of operations tells you which operation to perform first.
1-7 Simplifying Expressions Order of Operations
First: Perform operations inside grouping symbols. Second: Evaluate powers. Third: Fourth: Perform multiplication and division from left to right. Perform addition and subtraction from left to right.
1-7 Simplifying Expressions
Grouping symbols include parentheses ( ), brackets [ ], and braces { }. If an expression contains more than one set of grouping symbols, begin with the innermost set. Follow the order of operations within that set of grouping symbols and then work outward.
1-7 Simplifying Expressions Helpful Hint
Fraction bars, radical symbols, and absolute-value symbols can also be used as grouping symbols. Remember that a fraction bar indicates division.
1-7 Simplifying Expressions Additional Example 1: Simplifying Numerical Expressions Simplify each expression.
A. 15 – 2
3 + 1
15 – 15 – 2 6 3 + 1 + 1 9 + 1 10
B. 12 + 3 2
12 + 3 2
+ 10 ÷
+ 10 ÷ 2
2
There are no grouping symbols.
Multiply.
Subtract.
Add.
There are no grouping symbols.
12 + 9 + 10 ÷ 2
Evaluate powers. The exponent applies only to the 3.
12 + 9 + 26 5
Divide.
Add.
1-7 Simplifying Expressions Additional Example 1: Simplifying Numerical Expressions Simplify each expression.
C.
The fraction bar is a grouping symbol.
Evaluate powers. The exponent applies only to the 4.
Multiply above the bar and subtract below the bar.
Add above the bar and then divide.
1-7 Simplifying Expressions Check It Out!
Example 1a Simplify the expression.
There are no grouping symbols.
Rewrite division as multiplication.
Multiply.
48
1-7 Simplifying Expressions Check It Out!
Example 1b Simplify the expression.
3 7 21
The square root sign acts as a grouping symbol.
Subtract.
Take the square root.
Multiply.
1-7 Simplifying Expressions Check It Out!
Simplify the expression. Example 1c
The division bar acts as a grouping symbol.
Add and evaluate the power.
Multiply, subtract and simplify.
1-7 Simplifying Expressions Additional Example 2: Retail Application A shop offers gift-wrapping services at three price levels. The amount of money collected for wrapping gifts on a given day can be found using the expression 2B + 4S + 7D. On Friday the shop wrapped 10 basic packages B, 6 super packages S, and 5 deluxe packages D. Use the expression to find the amount of money collected for gift-wrapping on Friday.
2B + 4S +7D 2( 10 ) + 4( 6 ) + 7( 5 )
Substitute values for variables.
20 + 24 + 35 79
Multiply.
Add.
A total of $79 was collected on Friday.
1-7 Simplifying Expressions Check It Out!
Example 2 A formula for a player find Hank Aaron home runs.
’ ’ s total number of bases is Hits + D + 2T + 3H. Use this expression to s total bases for 1959, when he had 223 hits, 46 doubles, 7 triples, and 39
Hits + D + 2T + 3H 223 + 46 + 2( 7 ) + 3( 39 )
Substitute values for variables.
223 + 46 + 14 + 117
Multiply.
400
Add.
Hank Aaron ’ s total number of bases for 1959 was 400.
1-7 Simplifying Expressions
The terms of an expression are the parts to be added or subtracted. Like terms are terms that contain the same variables raised to the same powers. Constants are also like terms.
Like terms Constant
4x
–
3x
+
2
1-7 Simplifying Expressions
A coefficient is a number multiplied by a variable. Like terms can have different coefficients. A variable written without a coefficient has a coefficient of 1.
Coefficients 1
x
2 + 3
x
1-7 Simplifying Expressions
Like terms can be combined. To combine like terms, use the Distributive Property.
Distributive Property
ax
– bx = (a
–
b)x
Example
7x
–
4x = (7 – = 3x 4)x Notice that you can combine like terms by adding or subtracting the coefficients. Keep the variables and exponents the same.
1-7 Simplifying Expressions Additional Example 3: Combining Like Terms Simplify the expression by combining like terms.
A. 72p – 25p
72p – 25p
72p and 25p are like terms.
47p
Subtract the coefficients.
1-7 Simplifying Expressions Additional Example 3: Combining Like Terms Simplify the expression by combining like terms.
B.
A variable without a coefficient has a coefficient of 1.
and are like terms.
Write 1 as . Add the coefficients.
1-7 Simplifying Expressions Additional Example 3: Combining Like Terms Simplify the expression by combining like terms.
C. 0.5m + 2.5n
0.5
m
+ 2.5
n 0.5m and 2.5n are not like terms.
0.5m + 2.5n
Do not combine the terms.
1-7 Simplifying Expressions
Caution!
Add or subtract only the coefficients. 6.8y² – y² ≠ 6.8
1-7 Simplifying Expressions Check It Out!
Example 3 Simplify by combining like terms.
a. 16p + 84p
16p + 84p 100p
16p + 84p are like terms. Add the coefficients.
b. –20t – 8.5t
–20
t
– 8.5
t
–28.5t
c. 3m 2 + m 3 – m 2
3
m
2 2m 2 – m 2 + m 3 + m 3
20t and 8.5t are like terms.
Subtract the coefficients.
3m 2 and – m 2 are like terms.
Subtract coefficients.
1-7 Simplifying Expressions Additional Example 4: Simplifying Algebraic Expressions Use properties and operations to show that 14x + 4(2 + x) simplifies to 18x + 8.
1. Statements
14x + 4(2 + x)
Reasons 2.
14x + 4 (2) + 4 (x)
3.
14x + 8 + 4x
4.
14x + 4x + 8
5. 6.
( 14x + 4x 18x ) + 8 + 8 Distributive Property Multiply.
Commutative Property of Addition Associative Property of Addition Combine like terms.
1-7 Simplifying Expressions Check It Out!
Example 4 Use properties and operations to show that 6(x – 4) + 9 simplifies to 6x – 15. Statements Reasons 1. 2. 3. 4.
6(x
–
4) + 9 6
x
– 6x – 6 (4) + 9 24 + 9 6x – 15 Distributive Property Multiply.
Combine like terms.
1-7 Simplifying Expressions Lesson Quiz Simplify each expression.
1. 165 + 27 + 3 + 5 200
2.
8 3. The volume of a storage box can be found using the expression
l
w(w + 2). Find the volume of the box if
terms. l
= 3 feet and w = 2 feet.
24 ft 3
Simplify each expression by combining like 4.
5. 14c 2 – 9c 14c 2 – 9c 6. Use properties and operations to show that 24a + b ² + 3a + 2b ² simplifies to 27a + 3b ² .
Check students ’ work.