Transcript Simplifying Exponential Expressions

Regents Review #1
Expressions
Roslyn Middle School
Research Honors
Integrated Algebra
Simplifying Expressions
What does it mean to simplify an expression?
CARRY OUT ALL OPERATIONS!
Simplifying Exponential Expressions
1) xy0
2) (2x2y)(4xy3)
x(1)
8x3y4
24(x3)4(y5)4
16x12y20
x
any nonzero number
raised to the zero power
equals one
3) (2x3y5)4
multiply coefficients
add exponents
raise each factor to
to the power
Simplifying Exponential Expressions
2
4) 4 x y
xy2
4 xy
5)  x 
 y3 


1
4x
y
divide coefficients
subtract exponents
move negative exponents and
rewrite as positive
2
3
6)
x 6
9
y
9
y
x6
raise numerator and
denominator to the power
of the fraction
4 x3 y
24 xy3
1x 2 y 2
6
x2
2
6y
simplify numerator and
denominator coefficients by
dividing by a common factor
Simplifying Exponential Expressions
When simplifying exponential expressions, remember…
1) Use exponent rules to simplify
2) When dividing, all results appear in the numerator. If
negative exponents appear in the numerator, move
them to the denominator and rewrite them with
positive exponents.
3) Never ever allow a decimal to appear in the numerator
or denominator of your expression! All expressions
should have integer coefficients in the numerator and
denominator!!!
Scientific Notation
Writing numbers in scientific notation
1) 345,000,000 = 3.45  108
2) 0.0000109 = 1.09  10-5
Scientific Notation
Multiplying and Dividing Numbers in Scientific
Notation
3)
5
9
4) 5.6  10 12
4 10 3.2 10 
4  3.210 10 
9
5
12.8  10
4
1.28  10  10
1
1.28  10
3
4
4 10
12
5.6 10
 6
4 10
18
1.4 10
6
Polynomials
When adding polynomials, combine like terms!
1) (3x – 2) + (5x – y) + (2x – 4)
3x + 5x + 2x – 2 – 4 – y
10x – 6 – y
Polynomials
When subtracting polynomials, distribute the minus sign before
combining like terms!
2) Subtract 5x2 – 2y from 12x2 – 5
12x2 – 5y – (5x2 – 2y)
12x2 – 5y – 5x2 + 2y
12x2 – 5x2 – 5y + 2y
7x2 – 3y
Polynomials
When multiplying polynomials, distribute each
term from one set of parentheses to every term
in the other set of parentheses.
3) 3x  2x  4
3x 2  12 x  2 x  8
3x  10 x  8
2
4) x  2x  3x  2
3
2
2
x  3x  2 x  2 x  6 x  4
2
x3  5x 2  8x  4
Polynomials
When dividing polynomials, each term in the
numerator is divided by the monomial that
appears in the denominator.
5) 3 x 2 y 4  12 x 3 y 2
2
 3x
2
4
3
2
3x y
12 x y

2
2
 3x
 3x
4
2
 y  4 xy
Factoring
What does it mean to factor?
Factoring is the opposite of simplifying.
To factor means to create a product from a
simplified expression.
It is important to know how to factor because it
helps you simplify expressions!
Factoring
There are three ways to factor
1) Pull out the GCF  4 x 2  2 x  2 x(2 x  1)
2) AM factoring  x 2  5x  6  ( x  3)( x  2)
3) DOTS  9 x 2  16 y 4  (3x  4 y 2 )(3x  4 y 2 )
Factoring
When factoring completely, factor until you cannot
factor anymore!
1) 2 x  10 x  12
2
2)
4 x 2  36 y 2
2( x  5 x  6)
4( x 2  9 y 2 )
2( x  3)( x  2)
4( x  3 y )( x  3 y )
2
3)
 x2  x  2
 1( x 2  x  2)  pull out a  1
 1( x  2)( x  1)
Rational Expressions
When simplifying rational expressions (algebraic
fractions), factor and cancel out factors that are
common to both the numerator and
denominator.
1)
2)
3x  6
3( x  2)

 x2
3
3
x2  2x
x( x  2)
x


2
x  3x  2 ( x  2)( x  1) x  1
Rational Expressions
When multiplying, factor and cancel out common factors in the
numerators and denominators of the product.
3)
x 2  x  20
x
( x  5)( x  4)
x
( x  5)




x2  x
x2  2x  8
x( x  1)
( x  4)( x  2) ( x  1)( x  2)
When dividing, multiply by the reciprocal, then factor and cancel
out common factors in the numerators and denominators of the
product.
4)
x2  4
x2  6x  8
x2  4
x2 1


 2
2
x 1
x 1
x 1
x  6x  8
( x  2)( x  2)
( x  1)( x  1)
( x  2)( x  1)


( x  1)
( x  2)( x  4)
( x  4)
Rational Expressions
1) When adding and subtracting rational
expressions, find a common denominator.
2) Create equivalent fractions using the common
denominator(Multiply by FOOs)
3) Add or subtract numerators and keep the
denominator the same.
4) Simplify your final answer if possible.
Rational Expressions
4
2
2
5)

 LCD 9 x
2
9x
3x
x
4
2
3 Multiply by FOO
Multiply by FOO



2
x 9 x 3x
3
4x
6

2
2
9x
9x
4x  6
2( 2 x  3)

2
2
9 x simplified 9 x
Rational Expressions
6)
FOO
7x
x 1
7x
x 1
 2


 LCD ( x  4)( x  2)
x  2 x  2 x  8 x  2 ( x  4)( x  2)
x  4 7x
x 1


x  4 x  2 ( x  4)( x  2)
7 x ( x  4)
x 1
7 x 2  28 x  ( x  1)


( x  4)( x  2) ( x  4)( x  2)
( x  4)( x  2)
7 x 2  28 x  x  1 7 x 2  29 x  1

( x  4)( x  2)
( x  4)( x  2)
Radicals
When simplifying radicals, create a product
using the largest perfect square.
1)
48  16  3  4 3
When multiplying radicals, multiply coefficients
and multiply radicands.
2) 3 2 5 6   15 12  15  4  3  15  2  3  30 3
Radicals
When dividing radicals, divide coefficients and divide radicands.
3)
6 30
6

2
2 5
30
3
5
6
A fraction is not simplified, if a radical appears in the
denominator!
4)
3
2
3

2
3 2
2
2
 multiply by a foo
2
Radicals
When adding or subtracting radicals, simplify all radicals.
If radicals have “like” radicands, then add or subtract
coefficients and keep the radicands the same.
5)
2 32  4 18
2  16  2
24 2


4 3 2
8 2  12 2
4 2
4 9  2
In order to get like radicals,
simplify each radical.
Writing Algebraic Expressions
1) Express the cost of y shirts bought at x dollars each.
xy
2) Express the number of inches in f feet.
12f
Evaluating Algebraic Expressions
Evaluate x2 – y when x = -2 and y = -5
x2 – y
(-2)2 – (-5)  always put negative numbers in ( )
4+5
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Regents Review #1
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Using the information from this
power point and your review packet,
complete the practice problems.