Download Algebra IB Name Final Review Packet #1 Chapter 8: Powers

Algebra IB
Final Review Packet #1
Name ________________________________________
Chapter 8: Powers & Roots
An exponent tells how many times a number called the ________________ is used as a factor. Numbers that
are expressed using exponents are called _______________.
The multiplication expression
times. Using exponents
has a base of _________, 2 is used as a factor ___________
Write each expression using exponents.
1. 3*3*3*3 =
2. 5*5 =
3. 4 =
4. 6*6*6*6*6*6*6 =
5. 7*7*7 =
6. x*x*x*x =
7. 3*3*x*x*x*y =
8. 7*7*7*x*x*y*y*y =
9. 9*9*9*9*9*x*y*z*z =
10. (-5)*(-5)*(-5)*a*a*a*b*b =
Write each expression as a multiplication expression (i.e. 34= 3*3*3*3)
1. 32 =
5.
2. 41 =
6. (-2)4=
3. 73 =
7. 3322=
4. a4b5=
3x2y =
Evaluating Expressions
Example:
Evaluate the expression 4x 2 y , when x = 2 and y = 4 substitute the given values in for the variables –
4x 2 y = 4(2)2(4) = 4*2*2*4 = 64
Evaluate the following expressions when x = -2, y = 3, and z = -4
1. x5 =
6. -2(x3 + 1) =
2. 4y2 =
7. 2z2 =
3. -2z2 =
8. (-y)3 =
4. x2 – y2 =
9.
5. 3(y2+z) =
10. –x4 =
–y3 =
Multiplying Powers
When multiplying powers of the same base, _______________ the exponents, keeping the same base.
Simplify each expression:
1. 3x*x =
5. (3x3)(4x2) =
2. -4x4*x3=
6. (-5x2)(-2x4) =
3. 3x3*5x5 =
7. a3bc3 * ab7c4=
4. (a2b3)(a3b4) =
8. (-3c2)(2a2b4c2) =
Dividing Powers
When dividing powers of the same base, _______________ the exponents, keeping the same base.
Simplify each expression:
1.
53

52
2.
x4

x
3.
a 5b 3

a 2b 3
4.
25x 4 y 3

5x 2 y
5.
v 3 x 5u 9

v2 x 4u 5
6.
45x 5 y 6

9x 5
Negative Exponents
For a negative exponent: the expression a-n is the reciprocal of an.
1
This is written: a-n = .
a
Rewrite each expression with positive exponents.
1

3x 2
1. x 8 
2.
3. 6x 3 
4. x 3 y 4 
5.
1

(2x)3
6. (3x 2 )2 
Evaluate each expression.
1.
2.
3.
4.
When finding a power of a power; ________________ the exponents. For example, (x 3 )2  x 32  x 6
Simplify each expression.
1.(6xy)2 (x 2 y)3 
2. (4x 3 y 2 )3 (2x 2 y 4 ) 
3.(4z 4 )2 (2x 2 y)(3xy 3 z 5 ) 
4. -xy(-xy)2 
5. (-2x 3 y 3 z)4 (2xyz 4 )2 
Putting it all together: Simplify each expression using the properties of exponents. Use positive
exponents only.
2
 5
1.(6x)x 5 
2.   
 4
x6
3.

x3
4. 8x 2 y 5 z 3 
5.(3x 2 y)(8xy 4 ) 
6. (-x 2 y 4 z 5 )6 
7.
18x 2 y 2

6x 5 y 6
8.
15x 2 y 4 z 6

30x 1 y 2 z 8
Scientific Notation:
Scientific notation uses powers of ________ to write decimal numbers. Numbers written in scientific
notation contain a number between _________ and ________ multiplied by a power of ____________.
For example, the number 3.1 x 102, is in scientific notation. The number 45 x 102 is not in scientific
notation.
1. Write numbers in decimal form in scientific notation
a. Change 450 to scientific notation.
4.5x102 A number between ________ and _________ is needed. Move the decimal two places to the
left to get 4.5. Therefore, the power of 10 is a +2, since we moved the decimal two places to the
left.
b. Change .00987 to scientific notation.
9.87x10-3 A number between _______ and _______ is needed. Move the decimal three places to the
right to get 9.87. Therefore, the power of 10 is a -3, since we moved the decimal three places to
the right.
2. Rewrite numbers in scientific notation in decimal form.
a. Rewrite 3.4 x 103 in decimal form. 3,400. Since there is a positive power of 3, move the
decimal 3 places to the _________________.
b. Rewrite 1.87 x 10-5 in decimal form. .0000187. Since there is a negative power of 5,
move the decimal 5 places to the ____________________.
3. Multiply (5.4 x 103)(2.2 x 105)
(5.4 x 2.2)(103x105)
Regroup into decimals and powers of 10.
8
11.88(10 )
Simplify by multiplying decimals and adding powers
1.188 x 10 8+1
Add 1 to the 8 power since decimal moved 1 place to the left.
9
1.188 x 10
Rewrite each scientific notation in decimal form.
1. 2.08x10 5 
2. 4.5x10 3 
3. 6.25x10 6 
4. 9.5765x10 4 
Rewrite each decimal in scientific notation.
1. 68,000,000 =
2.
0.004953=
3. 1,490,000,000,000 =
4. 0.0975 =
Evaluate each expression in scientific notation.
1. (7x10 3 )(4x10 5 ) 
2. (5 x 10 5 )(3x10 2 ) 
3. (6x10 6 )(6x10 6 ) 
4. (7x10 6 )(5x10 1 ) 
Estimating Square Roots
You can estimate square roots by using perfect squares.
For example
6.
39 is estimated to be 6 since 39 is close to 36 a perfect square with the square root of
Estimate each square root to the nearest whole number.
1. 60
2.
250
3. 405
4.
520
5. 155
6.
90
7. Which is closer to
52 , 7 or 8?
8. Which is closer to 110 , 10 or 11?
The Pythagorean Theorem.
What does the Pythagorean Theorem tell us about the right triangle shown below?
Use the Pythagorean Theorem to solve for the missing side of the right triangles.
1.
2.
4. a = , b = 7, c = 100
5. a = 13, b = 2, c =
3.
6. a = 8, b = 24, c =
Chapter 9 – Polynomials
Define the following terms:
Monomial Some examples of monomials are - ______________________________________________________________
Polynomial –
Some examples of polynomials are - ______________________________________________________________
Binomial –
Some examples of binomials are - ______________________________________________________________
Trinomials –
Some examples of trinomials are - ______________________________________________________________
The degree of a monomial is the _________________________________________________________________
To find the degree of a polynomial, find the ____________________________________. The
_______________________________ of the degrees of its terms is the degree of the polynomial.
State the type (monomial, binomial, or trinomial) and the degree of each polynomial.
1.) 4  x
2.) p15  4 p
3.) 8r 2 s 2t 2
Type __________
Type __________
Type ___________
Degree________
Degree ________
Degree _________
4.) x  y  z
5.)
3xy  2yz  xyz 3
6.) 3pq 2  6q 2
Type __________
Type ____________
Type __________
Degree _________
Degree __________
Degree ________
Need more practice? Go to page page 386.
Adding & Subtracting Polynomials
Write a rule for adding polynomials:
Show three (3) examples of adding polynomials.
1.)
2.)
3.)
Write a rule for subtracting polynomials:
Show three (3) examples of subtracting polynomials.
1.)
2.)
3.)
Add or Subtract the following polynomials. (Need more practice? Go to page 392)
1. (4x2+6x-1)+(12-3x+5x2)
2.
(-6x2-x+3)+(10x2-5x-3)
3. (-x2-2x+1)+(-6x2+2x+1)
4. (13x2-4x-9)+(10-7x+6x2)
5. (-6x2+5x-1)+(13+3x-2x2)
6. (2x2+3x-y2)-(x2-3x+y2)
7.) (12x3-x2+7)-(7x3-x2+6)
8.) (4x2+14x-9)-(4x2+8x-7)
9.) (-2a3+5a2-1)-(-4a3+a2+2)
10.) (3a3-9a2-13)-(-8a3+a2+6)
Multiplying a polynomial by a monomial.
To multiply a polynomial by a monomial, use the ______________________________________.
Show three (3) examples of multiplying a polynomial by a monomial
1.)
2.)
3.)
Find each product.
1. -3x(2x2+3x-5)
2. 4y(y3-2y+1)
3. 2x(x2-2x-4)
4. -4x(3x+2y)-2x(x+3y)
5. x(x2-2x+4) – 2x(x2+4x-3)
6. –xy(2x-4y)
Multiplying Binomials
In section 9-4 you learned three different methods to multiply binomials.
Use the distributive property to multiply the following binomials:
(x+3)(x+6) =
Use the box method to multiply the following binomials:
(x+3)(x+6) =
Use the FOIL method to multiply the following binomials:
(x+3)(x+6) =
Multiply the following binomials using any method you like.
1. (x-6)(x-1) =
2. (x+7)(x+3) =
3. (x-12)(x-2) =
4.) (x+6)(x-14) =
5. (2y+1)(3y+4)
6. (3y-7)(2y+2) =
7.) (5a-2)(a+3) =
8.) (2y+5)(y-3)
9.) (2x-3)(2x+3) =
10.) (x-6)(x-6)
11.) (2x-3)(2x+3) =
12.) (3x+4)(3x+4) =
Chapter 10 – Factoring
FACTORING NUMBERS AND MONOMIALS
Factors are numbers you multiply together to get a product. Because 3 x 4 = 12 we say 3 and 4 are
____________________ of 12. Prime numbers are numbers that have exactly _________ factors, 1 and the
number itself (like 5). Composite numbers are numbers that have _________ factors (like 12).
List the factors of each number and classify the number as prime or composite.
1.
60
4.
23
2.
11
3.
5.
50
64
To completely factor monomials they are written as the product of their _____________ factors. To write
45 in factored form 45 = 3 * 3 *5. To list 12x2y in factored form ___________________________.
Factor each monomial.
1.
8x2
2. -4m2n
3.
-16x3y4
5. 28a3b4c
The Greatest Common Factor (GCF) of a set on monomials or a polynomial is the __________________
factor that the terms of a set of monomials or a polynomial have in common. To find the GCF you need
to factor the terms of a set of monomials or a polynomial and determine all _________________ they have in
common, then multiply those prime factors together.
For example, find the greatest common factor between 15ab2 & 12a2b2
15ab2 =
12a2b2 =
3*5*a*b*b
2*2*3*a*a*b*c
GCF = 3 * a * b * b = 3ab2
Find the greatest common factor (GCF) of each set of numbers or monomials.
1. 7ax, 21ax2, 35ax3
2.
3. 20, 24
6p, 3p2, 33px
4. 48x2y3z, 54xy2z, 36x3y3x3
What are the steps in factoring a polynomial by factoring out the GCF?
Factor each polynomial. If the polynomial cannot be factored, write prime.
1. 4x2 – 8x + 12
2. 6x3 – 15x2
4. 7x2 -21x
5. 9x6 + 81x3 – 27x
3. 7ab + 13c
FACTORING TRINOMIALS IN THE FORM: x2 + bx + c
When factoring a polynomial of the form x2 + bx + c (where the coefficient of your first term is 1) you
focus on the factors of the last term, determine the sign of your binomial factors, and then determine
which set of factors when combined will give you your middle term.
Factor:
x2 + 5x + 6
(x + )(x + )
(x +2)(x + 3)
The factors of 6 are _________________________
The sign of the last term is positive, the only way to get a positive
Product is to multiply (____, _____) or (______, ______)
The middle term of the polynomial is positive, so our binomial signs (___,___)
Find the factors of 6 that when added together give you 5
Remember, before finding the binomial factors of a polynomial check to see if the GCF has been
factored out. For example before trying to find the binomial factors of 4x2 + 24x + 36 make your life
easier by factoring out a 4!!!
You can always (and you should always) check your answer by using the foil method to see if when
you multiply your factors together you get the original polynomial!
Factor each polynomial.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.