Algebra I Notes Concept 00b: Review Properties of Integer Exponents

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1 Algera I Notes Concept 00: Review Properties of Integer Eponents In Algera I, a review of properties of integer eponents may e required. Students egin their eploration of power under the Common Core in Grade 6 y writing and evaluating epressions with eponents. In Grade, they etend their knowledge of power to properties of eponents. Specifically, students should eplore, learn, and apply eight properties of integer eponents in the real numers. Rational eponents will e addressed later in the course..ee. Know and apply the properties of integer eponents to generate equivalent numerical epressions. Knowledge Targets Students must know properties of eponents including: zero eponent, negative eponent, product/quotient of powers, power of power, power of product/quotient. The student will determine the value of eponential epressions using a variety of methods. The student will simplify algeraic epressions y applying the properties of eponents. Vocaulary ase eponent eponential form power Essential Questions / Big Ideas How can we rewrite and/or simplify epressions in eponential form? Knowing the properties of integer eponents allows us to rewrite epressions in equivalent form. Eponential Epression: consists of a ase and an eponent. The eponent tells you how many times to use the ase as a factor. a = a a a a a (a is used as a factor times) Mathematical Practices. Reason astractly and quantitatively.. Construct viale arguments and critique the reasoning of others. 7. Look for and make use of structure.. Look for and epress regularity in repeated reasoning. a is the BASE, is the EXPONENT a is the th POWER of a. Caution: There are some eponential epressions that can e tricky. E: Power of a Quantity: a a a a a a!!! SNRPDP Page of Alg I Concept 00 Notes Review Integer Eponents CCSS //0

2 Algera I Notes Concept 00: Review Properties of Integer Eponents E: Power of a Negative Numer: E: Negative Power of a Numer: Eploration: E: Simplify the epressions y epanding them first using the definition of an eponent a 6 7 a a a a a aa a a a a a a a a a 6 Using the pattern that emerged from the eamples aove, have students write a rule for multiplying eponential epressions. Product of Powers Property: To multiply powers with the same ase, add the eponents. E: Simplify the epressions using the product of powers property a a a a a a 6 Eploration: E: Simplify the epressions y epanding them first using the definition of an eponent. y. y y y y y y 6 0. SNRPDP Page of Alg I Concept 00 Notes Review Integer Eponents CCSS //0

3 Algera I Notes Concept 00: Review Properties of Integer Eponents Using the pattern that emerged from the eamples aove, have students write a rule for raising an eponential epression to a power. Power of a Power Property: To find a power of a power, multiply the eponents. Eploration: E: Simplify the epressions using the power of a power property. y.. y y y 6 0 E: Simplify the epressions y epanding them first using the definition of an eponent. y. a y y y y y y y y y y a a a a a a a 6. Using the pattern that emerged from the eamples aove, have students write a rule for raising a product to a power. Power of a Product Property: To find a power of a product, find the power of each factor and multiply. E: Simplify the epressions using the power of a product property. y y y y a a a. a 6. Comining the Properties y y. E: Simplify the epression Step One: Power of a product property y y SNRPDP Page of Alg I Concept 00 Notes Review Integer Eponents CCSS //0

4 Algera I Notes Concept 00: Review Properties of Integer Eponents 6 Step Two: Power of a power property 7 y y 7 y y Step Three: Product of powers property 7 y 7 y E: Simplify the epression a c c. Step One: Power of a product property a c c Step Two: Power of a power property a c6 c a c6 6 c Step Three: Product of powers property a c 6 c a c 6 6a c You Try: Simplify the epression using the properties of eponents. rs t r s QOD: Descrie the difference etween a and a, where a and are whole numers. Sample Practice Question(s): Evaluate when =. A. B. 6 C. 6 D. 6. Determine the value of A. B. C. 7 D... Which epression is equivalent to y z y? A. B. C. y z 7 y z 0 y z SNRPDP Page of Alg I Concept 00 Notes Review Integer Eponents CCSS //0

5 D. Algera I Notes Concept 00: Review Properties of Integer Eponents 7 y z SNRPDP Page of Alg I Concept 00 Notes Review Integer Eponents CCSS //0

6 Algera I Notes Concept 00: Review Properties of Integer Eponents Additional Information for Teachers Determine the value of eponential epressions using a variety of methods. Simplify algeraic epressions y applying the properties of eponents. Simplify epressions containing negative and zero eponents. Eploration: Have students use patterns to complete the tales. (Answers are in red.) 0 6 / / / /6 Note: The pattern is to divide y to move from left to right. 0 7 / / /7 / Note: The pattern is to divide y to move from left to right. Have students write a conjecture for the value of a ase raised to the zero or negative power. Zero Eponent: Any nonzero numer raised to the zero power is equal to a 0, a 0 Negative Eponent: For any nonzero ase, n a is the reciprocal of n a. a n n, a 0 a Evaluating Powers with Zero and Negative Eponents E: Evaluate the following Simplifying Eponential Epressions E: Simplify the epression. Write your answer with positive eponents. y. 0 a a 6. rs s. a. a 6. a c a SNRPDP Page 6 of Alg I Concept 00 Notes Review Integer Eponents CCSS //0

7 Algera I Notes Concept 00: Review Properties of Integer Eponents You Try: Simplify the epression eponents. QOD: Can you evaluate 0? Eplain. a. Write your answer with positive Sample Practice Questions: 0 Evaluate the epression. A. B. 0 C. D. 6. Evaluate the epression A. B. C. 0 D.. SNRPDP Page 7 of Alg I Concept 00 Notes Review Integer Eponents CCSS //0

8 Algera I Notes Concept 00: Review Properties of Integer Eponents Determine the value of eponential epressions using a variety of methods. Simplify algeraic epressions y applying the properties of eponents. Eploration: E: Simplify the epressions y epanding them first using the definition of an eponent Using the pattern that emerged from the eamples aove, have students write a rule for dividing powers that have the same ase. Quotient of Powers Property: To divide powers having the same ase, sutract the eponents. E: Simplify the epressions using the quotient of powers property Eploration: E: Simplify the epressions y epanding them first using the definition of an eponent.. a a a a a a a a a a a a a SNRPDP Page of Alg I Concept 00 Notes Review Integer Eponents CCSS //0

9 Algera I Notes Concept 00: Review Properties of Integer Eponents Using the pattern that emerged from the eamples aove, have students write a rule for dividing powers that have the same ase. Power of a Quotient Property: To find a power of a quotient, find the power of the numerator and the power of the denominator. E: Simplify the epressions using the power of a quotient property.. a a a Evaluating Powers Using the Division Properties of Eponents E: Evaluate the following epressions Simplifying Epressions with the Division Properties of Eponents E: Simplify the epressions. Write your answers with positive eponents. y 7 y. a a. a y You Try: Simplify the epression a a. QOD: Using the division property of eponents, show algeraically why and a 0. 0 a, when a is a real numer Sample Practice Question: t y y, what is the value of t? If y A. B. C. 6 D. SNRPDP Page of Alg I Concept 00 Notes Review Integer Eponents CCSS //0

10 Algera I Notes Concept 00: Review Properties of Integer Eponents Epress numers using scientific notation in mathematical and practical situations. Eploration: Find the product of the following epressions , ,,000, The numers used in the eploration were written in scientific notation. Write in your own words how to convert numers from scientific notation to decimal form. Converting from Scientific Notation to Decimal (Standard) Form: To convert from scientific notation with a positive power of 0, move the decimal point to the right. To convert from scientific notation with a negative power of 0, move the decimal point to the left. Note: Scientific notation is a short way to write long numers with many digits, whether they are very large or very small. A very large numer will have a positive power of 0, and a very small numer will have a negative power of 0. This should help you rememer which way to move the decimal. E: Convert the following to decimal form move decimal places to the right: 6. 0 move decimal places to the right:. 0 move decimal places to the left: 0 move decimal places to the left:.0 0, ,,000, Caution: Rememer that the eponent determines how many places to move the decimal point, NOT how many zeros are in the decimal form of the answer! Converting from Decimal Form to Scientific Notation: To convert to scientific notation, always move the decimal so that there is only one digit to the left of the decimal. Do not write any zeros, and use the numer of times you moved the decimal point as the power of 0. If it was a large numer (greater than ), use a positive power of 0, and if it was a small numer (smaller than ), use a negative power of 0. E: Convert the following to scientific notation. SNRPDP Page 0 of Alg I Concept 00 Notes Review Integer Eponents CCSS //0

11 Algera I Notes Concept 00: Review Properties of Integer Eponents,0,000,000 Step One: Move the decimal so that it is after the and drop the zeros. (Note: Right now, the decimal point is at the end of the numer.). Step Two: How many places did you need to move the decimal? Step Three: Write your answer in scientific notation using the numer found in Step Two as the power of 0. This was a large numer, so we will use positive as the eponent Step One: Move the decimal so that it is after the and drop the zeros..6 Step Two: How many places did you need to move the decimal? Step Three: Write your answer in scientific notation using the numer found in Step Two as the power of 0. This was a small numer, so we will use negative as the eponent Step One: Move the decimal so that it is after the and drop the zeros. (Note: Right now, the decimal point is at the end of the numer.) Step Two: How many places did you need to move the decimal? Step Three: Write your answer in scientific notation using the numer found in Step Two as the power of 0. This was a large numer, so we will use positive as the eponent Caution: This may appear to already e in scientific notation. However, ecause the decimal point is not to the right of just one digit, it is not. Step One: Write. in scientific notation. Move the decimal so that it is after the.. Step Two: How many places did you need to move the decimal? Step Three: Write your answer in scientific notation using the numer found in Step Two as the power of 0. This was a large numer, so we will use positive as the eponent.. 0 SNRPDP Page of Alg I Concept 00 Notes Review Integer Eponents CCSS //0

12 Algera I Notes Concept 00: Review Properties of Integer Eponents Step Four: Use the product property of eponents to simplify Multiplying and Dividing in Scientific Notation: Use the product and quotient properties of eponents to multiply and divide powers of 0. Be sure the final answer is in scientific notation. E: Find the product. 0 0 Step One: Use the associative property of multiplication. 0 0 Step Two: Simplify. Use the product of powers property of eponents E: Evaluate the epression. Step One: Rewrite as a product Step Two: Simplify. Use the quotient of powers property of eponents Step Three: Write the answer in scientific notation Finding Powers of Numers in Scientific Notation: Use the power of a product property and power of a power property of eponents to simplify. Write your final answer in scientific notation. E: Evaluate the epression..0 0 Step One: Use the power of a product property Step Two: Simplify. Use the power of a power property of eponents Step Three: Write the answer in scientific notation Scientific Notation on the Graphing Calculator SNRPDP Page of Alg I Concept 00 Notes Review Integer Eponents CCSS //0

13 Algera I Notes Concept 00: Review Properties of Integer Eponents E: Find the product of,00 and 00,000,000. The E on the calculator screen represents scientific notation. The numer after the E is the power of 0. Answer:. 0 Note: The calculator uses scientific notation ecause the numer of digits in the answer would not fit on the screen. E: Find the quotient: This time the calculator did not put the answer in scientific notation. This is ecause it had few enough digits to fit on screen. We will have to write the answer in scientific notation. the.00 0 E: Evaluate the power Write the answer in scientific notation Note: These are the same eamples we calculated y hand. Do you think it is easier to find products, quotients, and powers y hand in scientific notation or in decimal form? You Try: Evaluate the epression without a calculator. Write the result in scientific notation and in decimal form QOD: Eplain when and why scientific notation is used instead of the decimal form of a numer. Sample Practice Question(s): What is 7,00,000 in scientific notation? SNRPDP Page of Alg I Concept 00 Notes Review Integer Eponents CCSS //0

14 A. B. C. D. Algera I Notes Concept 00: Review Properties of Integer Eponents Multiply: What is the product in scientific notation? A. B. C. D SNRPDP Page of Alg I Concept 00 Notes Review Integer Eponents CCSS //0