Section 7.3 Rational Exponents Algebra 1

Learning Targets Define a rational exponent Define radical form Define and apply “ n th ” root Evaluate “ 𝑛 𝑡ℎ ” root expressions Define and apply Power Property of Equality Solve exponential equations

Recall: Exponent Definition 3 4 =3∙3∙3∙3=81 Remember, this is saying that I want 4 pieces of the base multiplied together

Multiplicative VS Additive Half 12 = 6+6 18=9+9 16=8+8 Multiplicative Half: 16=4∙4 36 = 6∙6 25= 5∙5 Thus, the additive half of 16 is 8 and the multiplicative half of 16 is 4.

Multiplicative VS Additive Thirds 9= 3+3+3 12 = 4+4+4 30=10+10+10 Multiplicative Third: 8= 2 ∙2 ∙2 27= 3 ∙3 ∙3 125= 5∙5∙5 Thus, the additive third of 30 is 10 and the multiplicative third of 125 is 5.

Explore: Exponent Definition What if I have something like… 81 1 4 This is saying that I know 81 is made of 4 equal pieces and I only want 1 of those pieces. Or I want the multiplicative fourth of 81 Thus, 81=3∙3∙3∙3 and one of the four pieces is just 3. So, 81 1 4 =3

Concept 1: Basic Rational Exponent The most common rational exponent is 𝑥 1 2 = 𝑥 𝑥 is also known as the square root. This representation is also known as radical form. 𝑥 1 2 is asking for the multiplicative half of a number 𝑥

Concept 1: Basic Rational Exponent Practice 1: What is 16 1 2 ? 16=4∙4 Thus, 16 1 2 =4 Practice 2: Find 100 100=10∙10 Thus, 100 =10

Concept 2: “ 𝒏 𝒕𝒉 ” Roots If 𝑎 𝑛 =𝑏, then 𝑏 1 𝑛 = 𝑛 𝑏 =𝑎 Ex: 2 4 =16, then 16 1 4 = 4 16 =2 4 16 = 16 1 4 . This is saying, I know 16 has 4 equal multiplicative pieces. It’s then asking, what is 1 of those 4 pieces. 2∙2∙2∙2=16 Thus, 16 1 4 =2

Concept 2: “ 𝒏 𝒕𝒉 ” Roots Practice 1: Practice 2: What is 27 1 3 ? 27=3∙3∙3 Thus, 27 1 3 =3 Practice 2: Find 5 32 32=2∙2∙2∙2∙2 Thus, 5 32 =2

Concept 2: “ 𝒏 𝒕𝒉 ” Roots Practice 3: Practice 4: What is 64 1 3 ? 64=4∙4∙4 Thus, 64 1 3 =4 Practice 4: Find 3 125 125=5∙5∙5 Thus, 3 125 =5

Concept 3: Advanced “ 𝒏 𝒕𝒉 ” Roots 𝑏 𝑚 𝑛 = 𝑛 𝑏 𝑚 Ex: 16 3 4 = 4 16 3 =8 4 16 3 = 16 3 4 . This is saying, I know 16 has 4 equal multiplicative pieces. It’s then asking, what is 3 of those 4 pieces. 2∙2∙2∙2=16 Thus, 16 3 4 =8

Concept 3: Advanced “ 𝒏 𝒕𝒉 ” Roots Practice 1: What is 27 2 3 ? 27=3∙3∙3 Thus, 27 2 3 =9 Practice 2: Find 36 3 2 36=6∙6 Thus, 36 3 2 =216

Concept 3: Advanced “ 𝒏 𝒕𝒉 ” Roots Practice 3: What is 64 2 3 ? 64=4∙4∙4 Thus, 64 2 3 =16 Practice 4: Find 32 2 5 32=2∙2∙2∙2∙2 Thus, 32 2 5 =4

Concept 4: Solving Exponential Equations Power Property of Equality For any real number 𝑏>0 and 𝑏≠1, then 𝑏 𝑥 = 𝑏 𝑦 if and only if 𝑥=𝑦. Example 1: If 5 𝑥 = 5 3 , then 𝑥=3 Example 2: If 2 𝑥+1 = 2 7 , then 𝑥+1=7

Concept 4: Solving Exponential Equations Practice 2: Solve 25 𝑥−1 =5 5 2 𝑥−1 = 5 1 2 𝑥−1 =1 2𝑥−2=1 Thus, 𝑥= 3 2 Practice 1: Solve 6 𝑥 =216 6 𝑥 = 6 3 Thus, 𝑥=3

Concept 4: Solving Exponential Equations Practice 4: Solve 12 2𝑥+3 =144 12 2𝑥+3 = 12 2 2𝑥+3=2 Thus, 𝑥=− 1 2 Practice 3: Solve 5 𝑥 =125 5 𝑥 = 5 3 Thus, 𝑥=3

Exit Ticket for Feedback 1. Solve 4 2𝑥−1 = 2 3 2. Find 8 2 3