Use Properties of Exponents Lesson 2.1 Honors Algebra 2 Use Properties of Exponents Lesson 2.1

Goals Goal Rubric Simplify expressions involving powers. Use scientific notation. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Scientific Notation

Powers & Exponents In an expression of the form an, a is the base, n is the exponent, and the quantity an is called a power. The exponent indicates the number of times that the base is used as a factor.

Powers & Exponents When the base includes more than one symbol, it is written in parentheses. A power includes a base and an exponent. The expression 23 is a power of 2. It is read “2 to the third power” or “2 cubed.” Reading Math

Properties of Exponents

Properties of Exponents

Simplifying Expressions The properties of exponents are used to simplify algebraic expressions. A simplified expression contains only positive exponents.

EXAMPLE 1 Evaluate numerical expressions a. (– 4 25)2 = (– 4)2 (25)2 Power of a product property = 16 25 2 Power of a power property = 16 210 = 16,384 Simplify and evaluate power. b. 115 118 –1 118 115 = Negative exponent property = 118 – 5 Quotient of powers property = 113 = 1331 Simplify and evaluate power.

Evaluate the expression. Your Turn: for Examples 1 Evaluate the expression. 1. (42 )3 2. (–8)(–8)3 4096 ANSWER Product of a powers property Power of a power property

Evaluate the expression. Your Turn: for Examples 1 Evaluate the expression. 3. 2 3 9 8 729 Power of a quotient property ANSWER 6 10 – 4 9 107 4. quotient of power property 2 3 1011 ANSWER Negative exponent property

a. b–4b6b7 = b–4 + 6 + 7 = b9 r–2 –3 s3 ( r – 2 )–3 ( s3 )–3 = = r 6 EXAMPLE 2 Simplify expressions a. b–4b6b7 = b–4 + 6 + 7 = b9 Product of powers property b. r–2 –3 s3 ( r – 2 )–3 ( s3 )–3 = Power of a quotient property = r 6 s–9 Power of a power property = r6s9 Negative exponent property c. 16m4n –5 2n–5 = 8m4n – 5 – (–5) Quotient of powers property = 8m4n0= 8m4 Zero exponent property

(x–3y3)2 (x–3)2(y3)2 x5y6 x5y6 x –6y6 x5y6 Power of a product property EXAMPLE 3 Standardized Test Practice SOLUTION (x–3y3)2 x5y6 = (x–3)2(y3)2 x5y6 Power of a product property x –6y6 x5y6 = Power of a power property

= x – 6 – 5y6 – 6 = x–11y 0 = x–11 1 x11 The correct answer is B. EXAMPLE 3 Standardized Test Practice = x – 6 – 5y6 – 6 Quotient of powers property = x–11y 0 Simplify exponents. = x–11 1 Zero exponent property = 1 x11 Negative exponent property The correct answer is B. ANSWER

EXAMPLE 4 Compare real-life volumes Betelgeuse is one of the stars found in the constellation Orion. Its radius is about 1500 times the radius of the sun. How many times as great as the sun’s volume is Betelgeuse’s volume? Astronomy

EXAMPLE 4 Compare real-life volumes SOLUTION Let r represent the sun’s radius. Then 1500r represents Betelgeuse’s radius. = 4 3 π (1500r)3 π r3 Betelgeuse’s volume Sun’s volume The volume of a sphere is πr3. 4 3 = 4 3 π 15003r3 π r3 Power of a product property

EXAMPLE 4 Compare real-life volumes = 15003r0 Quotient of powers property = 15003 1 Zero exponent property = 3,375,000,000 Evaluate power. Betelgeuse’s volume is about 3.4 billion times as great as the sun’s volume. ANSWER

Simplify the expression. Your Turn: for Examples 2, 3, and 4 Simplify the expression. 5. x–6x5 x3 Product of powers property x2 ANSWER 6. (7y2z5)(y–4z–1) 7z4 y2 Product of powers property Negative exponent property ANSWER

Simplify the expression. Your Turn: for Examples 2, 3, and 4 Simplify the expression. 7. s 3 2 t–4 s6t8 Power of a quotient property Power of a power property Negative exponent property ANSWER 8. x4y–2 3 x3y6 Quotient of powers property Power of a product property Power of a power property x3 y24 Negative exponent property ANSWER

Scientific Notation Scientific notation is a method of writing numbers by using powers of 10. In scientific notation, a number takes a form m  10n, where 1 ≤ m <10 and n is an integer. You can use the properties of exponents to calculate with numbers expressed in scientific notation.

EXAMPLE 5 Use scientific notation in real life A swarm of locusts may contain as many as 85 million locusts per square kilometer and cover an area of 1200 square kilometers. About how many locusts are in such a swarm? Locusts SOLUTION = 85,000,000 1200 Substitute values.

The number of locusts is about 1.02 1011, or about 102,000,000,000. EXAMPLE 5 Use scientific notation in real life = (8.5 107)(1.2 103) Write in scientific notation. = (8.5 1.2)(107 103) Use multiplication properties. = 10.2 1010 Product of powers property = 1.02 101 1010 Write 10.2 in scientific notation. = 1.02 1011 Product of powers property The number of locusts is about 1.02 1011, or about 102,000,000,000. ANSWER

Simplify the expression. Write the answer in scientific notation. Example 6 Simplify the expression. Write the answer in scientific notation. Divide 4.5 by 1.5 and subtract exponents: –5 – 6 = –11. 3.0  10–11

Simplify the expression. Write the answer in scientific notation. Your Turn: Simplify the expression. Write the answer in scientific notation. Solution Divide 2.325 by 9.3 and subtract exponents: 6 – 9 = –3. 0.25  10–3 2.5× 10 −1 × 10 −3 Because 0.25 < 10, move the decimal point right 1 place and subtract 1 from the exponent. 2.5  10–4

Simplify the expression. Write the answer in scientific notation. Your Turn: Simplify the expression. Write the answer in scientific notation. (4  10–6)(3.1  10–4) Solution (4)(3.1)  (10–6)(10–4) Multiply 4 by 3.1 and add exponents: –6 – 4 = –10. 12.4  10–10 1.24× 10 1 × 10 −10 Because 12.4 >10, move the decimal point left 1 place and add 1 to the exponent. 1.24  10–9