Lesson 3: Properties of Exponents
Lesson 3: Properties of Exponents
Lesson 3: Properties of Exponents
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Introductory Algebra<strong>Lesson</strong> 3 – <strong>Properties</strong> <strong>of</strong> <strong>Exponents</strong>Multiplication <strong>of</strong> PolynomialsMultiplication <strong>of</strong> MonomialsExample 1: Multiply and simplify.(3x 5 )( –2x 9 ) =Example 2: Expand and simplify.The Distributive Property5x 3 (2x 5 – 4x 3 – x + 8)=Example 3: Multiply and simplify.a. (x + 3)(x + 4) =Multiplication <strong>of</strong> Polynomialsb. (m – 5)(m – 6) =c. (2d – 4)(3d + 5) =d. (x – 2)(x 2 + 2x – 4) =
Introductory Algebra<strong>Lesson</strong> 3 – <strong>Properties</strong> <strong>of</strong> <strong>Exponents</strong>Example 4: Multiply and simplify.Squaring a Binomiala. (n + 5) 2 b. (3 – 2a) 22. Multiply and simplify.You Trya. –3x 2 (x 5 + 6x 3 – 5x)=b. (3x – 4)(5x + 2)=c. (2p – 5) 2 =
Introductory Algebra<strong>Lesson</strong> 3 – <strong>Properties</strong> <strong>of</strong> <strong>Exponents</strong>Applications from GeometryExample 1: Write a polynomial in simplest form that represents the area <strong>of</strong> the square.SOLUTION:The blue square has area:The yellow square has area:The pink rectangles each have area:Total Area ==Example 1 (another way): Write a polynomial in simplest form that represents the area <strong>of</strong> thesquare.SOLUTION:The total length <strong>of</strong> each side is x + y.Total Area = (x + y)( x + y)= x 2 + xy + yx + y 2Note that xy and yx are like terms: xy + yx = 2xyTotal Area = x 2 + 2xy + y 2Example 2: Write a polynomial in simplest form that represents the area <strong>of</strong> the shaded region.SOLUTION: To find the area <strong>of</strong> the shaded region we findthe area <strong>of</strong> the big square and subtract the area <strong>of</strong> the littlesquare.The big square has area:The little square has area:Area <strong>of</strong> the shaded region
Introductory Algebra<strong>Lesson</strong> 3 – <strong>Properties</strong> <strong>of</strong> <strong>Exponents</strong>You Try3. Write a polynomial in simplest form that represents the total area <strong>of</strong> the figure shown below.4. Write a polynomial in simplest form that represents the area <strong>of</strong> the dark blue region <strong>of</strong> thefigure shown below.
Introductory Algebra<strong>Lesson</strong> 3 – <strong>Properties</strong> <strong>of</strong> <strong>Exponents</strong>Division <strong>Properties</strong> <strong>of</strong> <strong>Exponents</strong>m nThe Division Property: a a 0aamnExample 1: Simplify the following expressions5010x4ab42x6ab5Raising a Quotient to a Power: b 0abnabnnExample 2: Simplify the following expressions2 541025 x4t73y6u5. Simplify the following expressions.You Trya.3a7102= b.3 86xy=59xy
Introductory Algebra<strong>Lesson</strong> 3 – <strong>Properties</strong> <strong>of</strong> <strong>Exponents</strong>
Introductory Algebra<strong>Lesson</strong> 3 – <strong>Properties</strong> <strong>of</strong> <strong>Exponents</strong>Negative <strong>Exponents</strong>For any real numbers a ≠ 0, b ≠ 0, and m:abmbamam1 1 mammaaExample 1: Rewrite each <strong>of</strong> the following with only positive exponents31a. x = b. =3xc.32 = d.452=e.43x = f.(3 )4x =Example 2: Simplify the following expressions.Write your answer with only positive exponents.a.4 2p p p b.2a35b3c2c.dd27= d.104tu=3 16t u
Introductory Algebra<strong>Lesson</strong> 3 – <strong>Properties</strong> <strong>of</strong> <strong>Exponents</strong>You Try6. Simplify the following expressions. Write your answers with only positive exponents7a.2ab.2 3 8n n n =c.46wx3wx2= d.2 32(3 x )
Introductory Algebra<strong>Lesson</strong> 3 – <strong>Properties</strong> <strong>of</strong> <strong>Exponents</strong>Division <strong>of</strong> PolynomialsSimplify the following expressions. Write your answer with only positive exponents.Example 1:6w30w83Example 2: 3 x 62Example 3:3 26x2x44xExample 4:220a35a45a2
Introductory Algebra<strong>Lesson</strong> 3 – <strong>Properties</strong> <strong>of</strong> <strong>Exponents</strong>You Try7. Simplify the following expressions. Write your answer with only positive exponents.a.11x153=b.23x5x123x2=
Introductory Algebra<strong>Lesson</strong> 3 – <strong>Properties</strong> <strong>of</strong> <strong>Exponents</strong>Scientific NotationScientific notation is the way that scientists easily handle very large numbers or very smallnumbers. For example, instead <strong>of</strong> writing 0.00000000000000092, we write 5.6 x 10 -16 .Powers <strong>of</strong> TenScientific Notation Standard Form10 4 10,000 3.21 x 10 4 = 32,10010 3 1,000 3.21 x 10 3 = 3,21010 2 100 3.21 x 10 2 = 32110 1 10 3.21 x 10 1 = 32.110 0 1 3.21 x 10 0 = 3.2110 -1 .1 3.21 x 10 -1 = 0.32110 -2 .01 3.21 x 10 -2 = 0.032110 -3 .001 3.21 x 10 -3 = 0.0032110 -4 .0001 3.21 x 10 -4 = 0.000321Writing Numbers in Scientific Notation and Standard FormScientific NotationStandard Form3.21 x 10 4 32,1003.21 x 10 –2 0.000321Example 1: Write the following numbers in standard form.a. 5.9 x 10 5 =b. 8.3 x 10 -7 =
Introductory Algebra<strong>Lesson</strong> 3 – <strong>Properties</strong> <strong>of</strong> <strong>Exponents</strong>Example 2: Write the following numbers in scientific notation.a. 8,140,000 =b. 0.0000000091 =On Your CalculatorExample 3: Evaluate the following on your calculator. Write in standard form.a. 850 6 =b. 0.25 8 =You Try8. Write the following numbers in standard form.a. 4.9 x 10 5 = b. 1.5 E -3 =9. Write the following numbers in scientific notation.a. 0.00000061 = b. 5,430,000,000 =
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