Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
Name/Period: _______________________________________
Date
Feb. 10 (B)
Feb. 12 (B)
Feb. 17 (B)
Feb. 19 (B)
Feb 23 (B)
Feb. 25 (B)
Feb. 29 (B)
Sect.
8.3
Unit Calendar
Topic
Exponents and Powers
Zero and Negative
Exponents
Homework
Page 506: 2-38, even
HW √
Essential Question: How can you simplify expressions using exponents?
8.1 Multiplying Powers with the Same Base
Page 492: 3-18
Essential Question: How can you simplify expressions using exponents?
8.2
Quiz #1
Multiplication and
Division Properties of
Exponents
Page 492: 19-37, odd
Page 499: 3-49, odd
On-Time
Essential Question: How can you simplify expressions using exponents?
8.4 Scientific Notation Page 515: 1-47, odd AND
Applications of
Scientific Notation
51 and 52
Essential Question: How can you use the graphing calculator to evaluate expressions
and equations using scientific notation?
2.7
11.2
Quiz #2
Radicals: Square roots
& cube roots
Real Number System pp. 723: 3-25
Express square roots and cube roots in simplest radical form
Essential Question: How do we express the square root/cube root of a function in its
simplest form?
Unit Review Review Packet
Study, study, study
Essential Question: How can we simplify expressions using exponents, scientific
notation, and square/cube roots?
Unit Test
Essential Question: How can we simplify expressions using exponents, scientific
notation, and square/cube roots?
Page 1 of 23

Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
Page 2 of 23

Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
Exponents & Powers
You can use powers to shorten how you represent repeated multiplication, such as:
2 x 2 x 2 x 2 x 2 x 2
2 6
A power has two parts, a base and an exponent. The exponent tells you how many times to use the base as a factor.
Examples.
5x 2 (5x) 3
What is the exponent?
What is the base?
You simplify a numerical expression when you replace it with its single numerical value. For example, the simplest form of 2 3 is 8 (2 x 2 x 2).
Examples. What is the simplified form of the expression?
10 7 = 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10
= 10,000,000
(0.2) 5 =
Practice. What is the simplified form of the expression?
1. 5 2 2. 2 5
3. 3 5 4. (½) 4
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Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
Zero and Negative Exponents
Properties of Zero as an Exponent: For every nonzero number, x, x 0 = 1 .
Examples. 4 0 = 1 (-3) 0 = 1 5.14
0 = 1
Properties of Negative Exponents: For every nonzero number, x, and integer, n, x -n = 1/x n
Examples. 7 −3 =
1
7
3
(−5) −2 =
1
(−5) 2
Note that this is specific to all numbers EXCEPT zero as the base. WHY???
Remember that an algebraic expression is in its simplest form when powers with a variable base
−5 0 are written only with positive exponents. Therefore, when simplifying expressions, we follow the
Properties of Zero and Negative Exponents.
Examples. What is the simplified form of each expression?
9 −2 = (−3.6) 0 = 4 −3 =
= (−4) −2 = 6 −1 =
5𝑎 3 𝑏 −2 =
1 𝑥 −5
= 𝑛
−5 𝑚 2
=
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Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
9𝑟
−2 𝑡
−3
3 2 𝑔 −4 ℎ 2
= 4𝑐 −3 𝑏 =
Practice. What is the simplified form of each expression?
(3/5 ) −1 =
1
8 0
=
5𝑔 −3 ℎ 4 =
2 −2 𝑥 −3 𝑦 −2
4 −2 𝑧 −3 𝑤 2
=
5𝑎 −2 𝑏 0 =
3 −4 =
6𝑎 −2 𝑏 −3
3𝑐 −5
=
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Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
Multiplying Powers with the Same Base
To multiply powers with the same base, add the exponents.
4
∙ 4
= 4
= 4
𝑏
∙ 𝑏
= 𝑏
= 𝑏
Let’s prove it….
3
∙ 3
Is the same as….
(3 ∙ 3 ∙ 3 ∙ 3) ∙ (3 ∙ 3)
Which is the same as….
= 𝑏
3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3
Which is the same as….
3
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Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
When variable factors have more than one base, be careful to combine only those powers with the same base.
When working with variables as the base, be sure to multiply the coefficients when simplifying the expression.
Examples. What is the simplified form of each expression?
12 𝟒 ∙ 12 𝟑 = 12 𝟒+𝟑 Add the exponents of the powers with the same base
= 12 𝟕 Simplify the exponent
= 358,318,808 Simplify
(−5) −2 ∙ (−5) 7
9 −3 ∙ 9 2 ∙ 9 6 =
=
4𝑧 5 ∙ 9𝑧 −12 = (4 ∙ 9)(𝑧 5 ∙ 𝑧 −12 ) Commutative and associative properties of multiplication
= 36(𝑧 5+(−12) ) Multiply the coefficients. Add the exponents of the powers with the same base.
= 36𝑧 −7
=
36 𝑧 7
Simplify the exponent.
Rewrite using a positive exponent.
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Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
2𝑎 ∙ 9𝑏 4
5𝑥 4 ∙ 𝑥 9
∙ 3𝑎 2 =
∙ 3𝑥 = 𝑗 2
−4𝑐 3
Practice. What is the simplified form of each expression?
−5 3 ∙ −5 −5 ∙ −5 −2 ∙ −5 5 =
∙ 7𝑑 2 ∙ 2𝑐 −2
∙ 𝑘 −2 ∙ 12𝑗 =
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Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
Multiplication & Division Properties of Exponents
You can use repeated multiplication to simplify a power raised to a power.
(x 5 ) 2 = x 5 ◦ x 5 = x 5 + 5 = x 10 ……………which is equivalent to x 5 ◦ 2 = x 10
Notice that (x 5 ) 2 = x 5 ◦ 2 . Raising a power is the same as raising the base to the product of the exponents.
Examples. What is the simplified form of the expression?
(n 4 ) 7 = y 3 (y 5 ) -2 =
(n 5 ) 2 (4mn -2 ) 3 =
(x -2 ) 2 (3xy 5 ) 4 =
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Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
Practice. What is the simplified form of the expression?
(p 5 ) 4 =
7(m 9 ) 3 =
(3g 4 ) -2 =
(x -2 ) 2 (3xy 5 ) 4 =
(6ab) 3 (5a -3 ) 2 =
Page 10 of 23

Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
To divide powers with the same base, subtract the exponents.
𝟓
𝟑
=
= 𝟒
𝟒
Examples. What is the simplified form of each expression? 𝑥 8
= 𝑥 6 𝑑 3 𝑑 9
= 𝑎 −3 𝑏 7
= 𝑎 5 𝑏 2
Practice. What is the simplified form of each expression? 𝑦 5
= 𝑦 4 𝑘 6 𝑗 2 𝑘𝑗 5
= 𝑥 4 𝑦 −1 𝑧 8
= 𝑥 4 𝑦 −5 𝑧
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Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
To raise a quotient to a power, raise the numerator AND the denominator to the power and
simplify.
Examples.
( 𝑎 𝑏
)
=
(
) 3 = 𝑎
𝑏
(
) 5 =
(
4
6
) -3 =
Practice.
3
10
= (
4
) 3 =
(
2
) -2 = (
4
) -4 =
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Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
Scientific Notation
Scientific notation is a shorthand way to write numbers using powers of 10. It is used to write very large or very small numbers more easily.
Why do you think it is called Scientific Notation?
A number in scientific notation is written as the product of two factors in the form of a × 10 n , where n is an integer and 1 < |a| < 10.
Examples of Scientific Notation.
8.3 × 10 3 4.12 × 10 22
Are these numbers in scientific notation?
0.23 × 10 -3 2.3 × 10 7
7.1 × 10 -22
9.3 × 100 9
With scientific notation, you use positive exponents to write numbers greater than 1. For example, 1,430,000,000 = 1.43 × 1,000,000,000 = 1.43 × 10 9 .
A shortcut to accomplish this is as follows:
You need to create your a which is between 1 and 10. Therefore, you need to move the decimal to the left to create an appropriate a. In this case, you move the decimal 9 places to the left.
1,430,000,000
Use the number of places that you used the decimal as your positive exponent to the base of 10. Again, in this case, you use 9 as your exponent.
10 9
Now you can write the number in scientific notation.
1.43 × 10 9
Page 13 of 23

Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
For numbers less between 0 and 1, you use negative exponents to write the number in scientific notation. For example, 0.0000000001 = 1 × 10 -10
A shortcut to accomplish this is as follows:
You need to create your a which is between 1 and 10. Therefore, you need to move the decimal to the right to create an appropriate a. In this case, you move the decimal 10 places to the right.
0.0000000001
Use the number of places that you used the decimal as your negative exponent to the base of 10. Again, in this case, you use -10 as your exponent.
10 -10
Now you can write the number in scientific notation.
1 × 10 -10
To convert numbers from scientific notation to standard notation, you do the opposite:
If the exponent is positive, move the decimal to the right.
5.5 × 10 6 = 5,500,000
If the exponent is negative, move the decimal to the left.
3.1 × 10 -3 = 0.0031
678,000 =
Practice. What is each number written in scientific notation?
0.000032 =
51,400,000 = 0.0000007 =
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Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
What is each number written in standard notation?
5.23 × 10 7 = 4.6 × 10 -5 =
2.09 × 10 -4 = 3.8 × 10 12 =
Comparing and Ordering Numbers in Scientific Notation
To compare and order number in scientific notation:
1.
Compare the powers of 10.
2.
Order the numbers based on the powers. a.
If ordering least to greatest, order the powers least to greatest. b.
If ordering greatest to least, order the powers greatest to least.
3.
If numbers have the same power of 10, then compare the decimals. a.
If ordering least to greatest, order the decimals least to greatest. b.
If ordering greatest to least, order the decimals greatest to least.
Example.
The earth’s four major oceans with their corresponding surface areas are:
Arctic = 1.41× 10 7 , Atlantic = 1.06 × 10 8 , Indian = 7.49 × 10 7 , and Pacific = 1.8 × 10 8 .
What is the order of the oceans from least to greatest surface area?
Practice. What is the order of the following parts of an atom from least to greatest mass? neutron: 1.675 × 10 -24 g; electron: 9.109 × 10 -28 g, proton: 1.673 × 10 -24 g
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Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
Using the Graphing Calculator for Scientific Notation
You can use your graphing calculator to work with numbers in scientific notation. The E on the calculator’s readout stands for exponentiation. The readout 1.35
E 8 means 1.35 × 10 8 , or
135,000,000. The EE key lets you input an exponent for a power of 10. So to enter 4 × 10 6 , you enter 4 EE 6.
You can work with your calculator in either Normal Mode or Scientific Mode to work with scientific notation. To change the mode on the calculator, select MODE and move the cursor to the right or left to choose either NORMAL or SCI.
In Normal Mode: If a problem is already expressed in scientific notation, you may simply enter the problem on the home screen in Normal Mode to arrive at the answer. Note that the answer will be displayed in standard format unless the answer cannot be displayed in 10 digits or if the absolute value is less than 0.001.
Example. With the calculator in Normal Mode, enter 4.726 × 10 -3 .
4.726 *10^-3 OR 4.726 EE-3
Now enter 4.726 × 10 -12 . What is displayed this time? What does the
E
mean?
In Scientific Mode: If a problem asks for the answer to be expressed in scientific notation, you will want to change the Mode to SCI and your answers will be displayed in scientific notation, regardless of the number of decimals.
Example. With the calculator in Normal Mode, enter 4.726 × 10 -3 .
4.726 *10^-3 OR 4.726 EE-3
E
Now enter 9.3 × 10 7 . What is displayed this time?
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Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
Calculations Using Scientific Notation
The properties of exponents are used when calculating with numbers written in scientific notation.
Examples. Write answers in scientific notation.
(2.5 × 10 -3 ) ∙ (3.0 × 10 5 ) (6.3 × 10 5 ) ∙ (1.2 × 10 3 )
2.5 ∙ 3.0 × 10 -3 ∙ 10 5
7.5 × 10 2
8.4 × 10
9
2.0 × 10 5
8.4
2.0
10
9
10 5
= 4.2 × 10 4
(1.5 × 10 5 ) ÷ (8.0 × 10 −2 )
(2.1 × 10 4 ) 2
(2.1) 2 × (10 4 ) 2
4.41 × 10 8
(5.0 × 10 6 ) 3
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Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
Practical Problems Using Scientific Notation
Write out the steps you need to solve each problem and then solve. Using a graphing calculator is
highly recommended.
1) The mass of one molecule of water is 2.99 x 10 -23 g. If a cylinder contains 3.0 x 10 50 molecules of water, what is the mass of the water in the container?
2) The diameter of Earth is 1.2756 x 10 4 km and the diameter of our sun is 1.39 x 10 6 km.
What is the ratio of the diameter of earth to the diameter of our sun?
3) The average distance from the sun to Mars is 2.28 x 10 8 km. The speed of light is 3 x 10 5 km/s. How long does it take for light from the sun to reach Mars?
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Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
4) The mass of Mercury is 3.3 x 10 23 . The mass of Venus is 4.87 x 10 24 . The mass of Earth is
5.98 x 10 24 . What is the combined mass of these three planets?
5) The average distance from the sun to Jupiter is 7.78 x 10 8 km. The average distance from the sun to Saturn is 1.43 x 10 9 km. How much farther from the sun is Saturn than Jupiter?
6) California has an area of approximately 1.56 x 10 5 square miles. California has a population of about 2.98 x 10 7 . How many people are there per square mile?
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Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
Squares, Cubes and Roots
Perfect squares are the product of multiplying two identical numbers – taking a number to the power of 2 (x 2 ). Fill in the table of perfect squares: x 1 2 3 4 5 6 7 8 9 10 11 12 x 2
√64
The square root of a number is one of the two identical factors of the number.
{Identical Twins!}
Positive numbers have a positive square root and a negative square root.
√16 √25 √125
Negative numbers do not have a real square roots because the square of a number is never negative.
√−4
Practice.
√−81 √−169
−√121 √−100
Page 20 of 23

Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
Taking a number to the 3 rd power (x 3 ) is called finding the cube of that number. Fill in the table of perfect cubes. x 1 2 3 4 5 6 7 8 9 10 x 3
The cube root of a number is one of the three identical factors of the number.
{Identical Triplets!}
Positive numbers have positive cube roots.
3
√9
3
√125
3
√64
Negative numbers have negative cube roots.
3
√−1000
3
√−64
Practice.
3
√−27
3
√216
3
√−8
3
− √−9
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Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
When simplifying radical expressions, we look for identical twins (square roots) and identical
triplets (cube roots). After identifying all identical items, any “left-overs” remain under the radical.
Examples.
√18 √4𝑥 2 𝑦 5 √625
3
√54
Practice.
3
√81
3
√9𝑥 7 𝑦 6
3
√75𝑟 9 √28𝑥 3 𝑦 3 √100𝑣 3
3
√250𝑗 3 𝑘 6 3
√192𝑥 √180
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Briggs – Algebra 1 – Unit 7: Polynomial Operations & Radicals -- Notes
All Properties of Exponents apply only to non-zero bases.
Zero Exponent Property
Negative Exponent Property
Property of Powers Property
Quotient of Powers Property
Power of a Product Property
Power of a Quotient Property
Power of a Power Property 𝑎 𝑎 𝑎
−𝑏
1
0
−𝑏
= 1
=
1 𝑎 𝑏
= 𝑎 𝑏 𝑎 𝑏 ∙ 𝑎 𝑐 = 𝑎 𝑏+𝑐 𝑎 𝑎 𝑏 𝑐
(𝑎𝑏)
(
(𝑎 𝑎 𝑏 𝑏
)
) 𝑐
= 𝑎 𝑏−𝑐 𝑐 𝑐
= 𝑎 𝑐
= 𝑎 𝑐 𝑏 𝑐
= 𝑎 𝑏 𝑐 𝑏𝑐
Any base with the power of 0 is equal to 1.
A negative exponent means that you create the multiplicative inverse of the base with a positive exponent.
When you multiply two powers with the same base, you add the exponents.
When you divide two powers with the same base, you subtract the exponents.
To find the power of a product, either find the power of each factor and then multiply or multiply the factors and raise the product to the power.
To find the power of a quotient, either find the power of each factor and then divide or divide the factors and raise the quotient to the power.
To find a power of a power, multiply the exponents.
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