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**4.4 Fractional Exponents and Radicals**

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**Construct Understanding**

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**Construct Understanding**

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REMEMBER Grade 9? 𝒂 𝒎 BASE

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REMEMBER Grade 9? 𝒂 𝒎 EXPONENT

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**𝒂 𝒎 • 𝒂 𝒏 = 𝒂 𝒎+𝒏 REMEMBER Grade 9?**

𝒂 𝒎 • 𝒂 𝒏 = 𝒂 𝒎+𝒏 We can further use it to calculate fractional exponents with numerator 1…

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**WHAT IS A FRACTIONAL EXPONENT?**

𝒂 x y

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**A FRACTIONAL EXPONENT with a numerator 1**

𝒂 1 y

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**Without a calculator, Calculate**

𝟓 𝟏 𝟐 • 𝟓 𝟏 𝟐 = 𝟓 𝟏 𝟐 + 𝟏 𝟐 = 𝟓 𝟐 𝟐 = 𝟓 𝟓 • 𝟓 = 𝟐𝟓 = 𝟓 What do you notice?

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**𝟓 𝟏 𝟐 and 𝟓 equivalent expressions**

𝟓 𝟏 𝟐 = 𝟓 𝟓 𝟏 𝟐 • 𝟓 𝟏 𝟐 = 𝟓 𝟏 𝟐 + 𝟏 𝟐 = 𝟓 𝟐 𝟐 = 𝟓 𝟓 • 𝟓 = 𝟐𝟓 = 𝟓 𝟓 𝟏 𝟐 and 𝟓 equivalent expressions

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𝟓 𝟏 𝟐 = 𝟓 𝟓 𝟏 𝟐 • 𝟓 𝟏 𝟐 = 𝟓 𝟏 𝟐 + 𝟏 𝟐 = 𝟓 𝟐 𝟐 = 𝟓 𝟓 • 𝟓 = 𝟐𝟓 = 𝟓 Similarly,

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**Without a calculator, Calculate**

𝟓 𝟏 𝟑 · 𝟓 𝟏 𝟑 · 𝟓 𝟏 𝟑 = 𝟓 𝟏 𝟑 + 𝟏 𝟑 + 𝟏 𝟑 = 𝟓 𝟑 𝟑 = 𝟓 𝟑 𝟓 · 𝟑 𝟓 · 𝟑 𝟓 = 𝟑 𝟏𝟐𝟓 = 𝟓 What do you notice?

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**𝟓 𝟏 𝟑 and 𝟑 𝟓 equivalent expressions**

𝟓 𝟏 𝟑 = 𝟑 𝟓 𝟓 𝟏 𝟑 · 𝟓 𝟏 𝟑 · 𝟓 𝟏 𝟑 = 𝟓 𝟏 𝟑 + 𝟏 𝟑 + 𝟏 𝟑 = 𝟓 𝟑 𝟑 = 𝟓 𝟑 𝟓 · 𝟑 𝟓 · 𝟑 𝟓 = 𝟑 𝟏𝟐𝟓 = 𝟓 𝟓 𝟏 𝟑 and 𝟑 𝟓 equivalent expressions

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**𝟓 𝟏 𝟐 and 𝟓 equivalent expressions**

𝟓 𝟏 𝟐 = 𝟓 𝟓 𝟏 𝟐 • 𝟓 𝟏 𝟐 = 𝟓 𝟏 𝟐 + 𝟏 𝟐 = 𝟓 𝟐 𝟐 = 𝟓 𝟓 • 𝟓 = 𝟐𝟓 = 𝟓 𝟓 𝟏 𝟐 and 𝟓 equivalent expressions

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This suggests 𝟓 𝟏 𝟐 = 𝟓 𝟓 𝟏 𝟑 = 𝟑 𝟓 𝒙 𝟏 𝒏 = 𝒏 𝒙

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**Powers with Rational Exponents**

𝟓 𝟏 𝟐 = 𝟓 𝟓 𝟏 𝟑 = 𝟑 𝟓 𝒙 𝟏 𝒏 = 𝒏 𝒙 When n is a natural number and x is a rational number,

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**Evaluate each power without using a calculator**

𝟐𝟕 𝟏 𝟑 = 3 27 = 3 𝟎.𝟒𝟗 𝟏 𝟐 = 0.49 = 0.7 ( 𝟒 𝟗 ) 𝟏 𝟐 = 𝟒 𝟗 = 𝟐 𝟑

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**POWERPOINT PRACTICE PROBLEM Evaluate each power without using a calculator**

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**The numerator in the exponent IS NOT 1?**

What if…. The numerator in the exponent IS NOT 1? 𝟖 𝟏 𝟑 𝟖 𝟐 𝟑 𝒙 𝟏 𝒏 = 𝒏 𝒙 ??

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RECALL.. (𝒂 𝒎 ) 𝒏 = 𝒂 𝒎 • 𝒏 So, for example,

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𝟖 𝟐 𝟑 = 𝟖 𝟏 𝟑 · 𝟐 (𝒂 𝒎 ) 𝒏 = 𝒂 𝒎 • 𝒏 𝒂 𝒎 • 𝒏 = (𝒂 𝒎 ) 𝒏 = ( 𝟖 𝟏 𝟑 ) 𝟐 But, this is also true… = ( 𝟑 𝟖 ) 𝟐 = (𝟐) 𝟐 = 4

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𝟖 𝟐 𝟑 = 𝟖 𝟐 · 𝟏 𝟑 (𝒂 𝒎 ) 𝒏 = 𝒂 𝒎 • 𝒏 𝒂 𝒎 • 𝒏 = (𝒂 𝒎 ) 𝒏 = ( 𝟖 𝟐 ) 𝟏 𝟑 = 𝟑 𝟖 𝟐 But, this is also true… = 𝟑 𝟔𝟒 𝒙 𝟏 𝒏 = 𝒏 𝒙 = 4

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**Powers with Rational Exponents**

When m and n are natural numbers and x is a rational number, 𝒙 𝒎 𝒏 = ( 𝒙 𝟏 𝒏 ) 𝒎 = ( 𝒏 𝒙 ) 𝒎 𝒙 𝒎 𝒏 = ( 𝒙 𝒎 ) 𝟏 𝒏 = 𝒏 𝒙 𝒎 AND

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**Write 𝟒𝟎 𝟐 𝟑 in radical form in 2 ways**

Write 𝟑 𝟓 and ( 𝟑 𝟐𝟓 ) 𝟐 in exponent form. 𝟒𝟎 𝟐 𝟑 = ( 𝟑 𝟒𝟎 )² and 𝟑 𝟒𝟎² 𝟑 𝟓 = 𝟑 𝟓 𝟐 ( 𝟑 𝟐𝟓 ) 𝟐 = 𝟐𝟓 𝟐 𝟑 𝒙 𝒎 𝒏 = ( 𝒙 𝟏 𝒏 ) 𝒎 = ( 𝒏 𝒙 ) 𝒎 𝒙 𝒎 𝒏 = ( 𝒙 𝒎 ) 𝟏 𝒏 = 𝒏 𝒙 𝒎

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POWERPOINT PRACTICE PROBLEM a) Write 𝟐𝟔 𝟐 𝟓 in radical form in 2 ways b) Write 𝟔 𝟓 and ( 𝟒 𝟏𝟗 ) 𝟑 in exponent form.

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**A husky with a body mass of 27 kg **

Biologists use the formula b = 0.01 𝒎 𝟐 𝟑 to estimate the brain mass , b kilograms, of a mammal with body mass m kilograms. Estimate the brain mass each animal A husky with a body mass of 27 kg A polar bear with a body mass of 200g

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**Biologists use the formula b = 0**

Biologists use the formula b = 0.01 𝒎 𝟐 𝟑 to estimate the brain mass , b kilograms, of a mammal with body mass m kilograms. Estimate the brain mass each animal A husky with a body mass of 27 kg A polar bear with a body mass of 200g Substitute: m = 27 b = 0.01 (𝟐𝟕) 𝟐 𝟑 b = 0.01 (∛𝟐𝟕) 𝟐 b = 0.01 (𝟑) 𝟐 b = 0.09 kg The brain mass of the husky is approximately 0.09 kg.

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**b = 0.01 (𝟐𝟎𝟎) 𝟐 𝟑 USE a CALCULATOR! Substitute: m = 200**

Biologists use the formula b = 0.01 𝒎 𝟐 𝟑 to estimate the brain mass , b kilograms, of a mammal with body mass m kilograms. Estimate the brain mass each animal A husky with a body mass of 27 kg A polar bear with a body mass of 200g Substitute: m = 200 b = 0.01 (𝟐𝟎𝟎) 𝟐 𝟑 USE a CALCULATOR! The brain mass of the polar bear is approximately 0.34 kg.

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**POWERPOINT PRACTICE PROBLEM Use the formula b = 0**

POWERPOINT PRACTICE PROBLEM Use the formula b = 0.01 𝒎 𝟐 𝟑 to estimate the brain mass of each animal. A moose with a body mass of 512 kg A cat with a body mass of 5 kg

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Section 4.4 HOMEWORK PAGES: 227 – 228 PROBLEMS:

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QUIZ Sections 4.1 – 4.3 Friday, July 25

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