Bell Ringer 1. (𝟐 𝒂 𝟑 )(𝟓 𝒂 𝟒 ) A.𝒙 2. 𝟑 𝒙 −𝟒 𝒚 𝟑 𝟐 B. 𝟖𝒙 𝟏𝟐 𝟐𝟕 Match each word in Column A with the matching answer of Column B. Column A Column B 1. (𝟐 𝒂 𝟑 )(𝟓 𝒂 𝟒 ) 2. 𝟑 𝒙 −𝟒 𝒚 𝟑 𝟐 3. 𝟒 𝒂 𝟖 𝟐 𝒂 𝟒 4. 𝟐 𝒙 𝟑 𝒚 𝟕 −𝟐 5. 𝟏 𝒙 −𝟐 −𝟏 ∙ 𝒙 𝟑 6. 𝟐𝒙 𝟒 𝟑 𝟑 A.𝒙 B. 𝟖𝒙 𝟏𝟐 𝟐𝟕 C. 𝟏𝟎 𝒂 𝟕 D. 𝟗 𝒚 𝟔 𝒙 𝟖 E. 𝟏 𝟒 𝒙 𝟔 𝒚 𝟏𝟒 F. 𝟐 𝒂 𝟒

Bell Ringer Mrs. Rivas 𝒙+𝒚 𝟐 =𝟒𝟎 𝒙 𝟐 +𝟐𝒙𝒚+ 𝒚 𝟐 =𝟒𝟎 𝒙 𝟐 +𝟐𝒙𝒚+ 𝒚 𝟐 =𝟒𝟎 ISCHS Bell Ringer Suppose 𝒙𝒚=𝟗 and 𝒙+𝒚 𝟐 =𝟒𝟎, what is 𝒙 𝟐 + 𝒚 𝟐 ?. 𝒙+𝒚 𝟐 =𝟒𝟎 𝒙 𝟐 +𝟐𝒙𝒚+ 𝒚 𝟐 =𝟒𝟎 𝒙 𝟐 +𝟐𝒙𝒚+ 𝒚 𝟐 =𝟒𝟎 𝒙 𝟐 +𝟐(𝟗)+ 𝒚 𝟐 =𝟒𝟎 𝒙 𝟐 +𝟏𝟖+ 𝒚 𝟐 =𝟒𝟎 −𝟏𝟖 −𝟏𝟖 𝒙 𝟐 + 𝒚 𝟐 =𝟐𝟐

Roots and Radical Expressions. Mrs. Rivas ISCHS Section 6-1 Roots and Radical Expressions. Objective: To find nth roots. Corresponding to every power there is a root. Example: 5 is a square root of 25. 𝟓 𝟐 =𝟐𝟓 5 is a cube root of 125. 𝟓 𝟑 =𝟏𝟐𝟓 5 is a fourth root of 625. 𝟓 𝟒 =𝟔𝟐𝟓 5 is a fifth root of 3125. 𝟓 𝟓 =𝟑𝟏𝟐𝟓 This pattern suggests a definition of an nth root.

Roots and Radical Expressions. Mrs. Rivas ISCHS Section 6-1 Roots and Radical Expressions. 𝒂 𝒏 =𝒃 𝑰𝒇 𝒏 𝒊𝒔 𝒐𝒅𝒅… 𝑰𝒇 𝒏 𝒊𝒔 𝑬𝒗𝒆𝒏… there is 𝐨𝐧𝐞 𝐫𝐞𝐚𝐥 𝒏𝒕𝒉 root of 𝒃, denoted in radical form as 𝑛 𝑏 . and 𝒃 𝐢𝐬 𝐩𝐨𝐬𝐢𝐭𝐢𝐯𝐞, there are two real nth roots of b. The positive root is the PRINCIPAL ROOT and its symbol is 𝑛 𝑏 . The negative root is its opposite, or − 𝑛 𝑏 . 𝒐𝒅𝒅 𝒃 = 𝟑 𝟐𝟕 =𝟑 𝒆𝒗𝒆𝒏 𝒃 = 𝟐 𝟏𝟔 =𝟒 𝒐𝒅𝒅 −𝒃 = 𝟑 −𝟐𝟕 =−𝟑 and b is negative, there are NO real nth roots of b. The only nth root of 0 is 0. 𝒆𝒗𝒆𝒏 −𝒃 = 𝟐 −𝟏𝟔 =𝑵𝒐 𝒓𝒆𝒂𝒍 𝒓𝒐𝒐𝒕

Roots and Radical Expressions. Mrs. Rivas ISCHS Section 6-1 Roots and Radical Expressions. 𝒏 𝒂 𝒏 𝒏 𝒂 𝒏 Means: You must Include the absolute value 𝒂 when n is EVEN. 𝒙 𝟒 𝒚 𝟔 = 𝒙 𝟐 𝒚 𝟑 𝟐 = 𝒙 𝟐 𝒚 𝟑 = 𝒙 𝟐 𝒚 𝟑 You must Omit the absolute value when n is ODD. 𝟑 𝒙 𝟑 𝒚 𝟔 = 𝟑 𝒙 𝒚 𝟐 𝟑 =𝒙 𝒚 𝟐

Roots and Radical Expressions. Mrs. Rivas ISCHS Section 6-1 Roots and Radical Expressions. Example # 1 Simplifying radical expressions. What is a simpler form of each radical expression? A 𝟏𝟔 𝒙 𝟖 The index 2 is even, USE absolute value symbols 𝟏𝟔 𝒙 𝟖 = 𝟒 2 𝒙 𝟒 2 |4 𝑥 4 | = 4 𝑥 4 because 𝑥 4 is never negative. = 𝟒𝒙 𝟒 2 = 𝟒𝒙 𝟒 =𝟒 𝒙 𝟒

The index 3 is ODD, DO NOT USE absolute value symbols Mrs. Rivas ISCHS Section 6-1 Roots and Radical Expressions. Example # 1 Simplifying radical expressions. What is a simpler form of each radical expression? B 𝟑 𝒂 𝟔 𝒃 𝟗 The index 3 is ODD, DO NOT USE absolute value symbols 𝟑 𝒂 𝟔 𝒃 𝟗 = 𝟑 𝒂 𝟔 𝒃 𝟗 𝟑 = 𝟑 𝒂 𝟐 𝟑 𝒃 𝟑 𝟑 = 𝒂 𝟐 𝒃 𝟑

Roots and Radical Expressions. Mrs. Rivas ISCHS Section 6-1 Roots and Radical Expressions. You Do It Simplifying radical expressions. What is a simpler form of each radical expression? C 𝟒 𝒙 𝟏𝟐 𝒚 𝟏𝟔 A 𝟖𝟏 𝒙 𝟒 B 𝟑 𝒂 𝟏𝟐 𝒃 𝟏𝟓 = 𝟗𝒙 𝟐 = 𝒂 𝟒 𝒃 𝟓 = 𝒙 𝟑 𝒚 𝟒

Mrs. Rivas ISCHS Section 6-2 Multiplying and Dividing Radical Expressions. Objective: To multiply radical expressions 𝑛 𝑎 ∙ 𝒏 𝑏 = 𝒏 𝑎𝑏

𝟓 𝐲 Mrs. Rivas Simplify a product. = 𝟏𝟐 𝒙 𝟒 𝒚 𝟐 𝟏𝟓 𝒙 𝟐 𝒚 𝟑 ISCHS Section 6-2 Multiplying and Dividing Radical Expressions. Example # 1 Simplify a product. What is the simplest form of 12 𝑥 4 𝑦 2 ∙ 15 𝑥 2 𝑦 3 ? = 𝟏𝟐 𝒙 𝟒 𝒚 𝟐 𝟏𝟓 𝒙 𝟐 𝒚 𝟑 = 𝟏𝟐∙𝟏𝟓 𝒙 𝟒+𝟐 𝒚 𝟐+𝟑 = 𝟐∙𝟐∙𝟑∙𝟑∙𝟓∙𝒙∙𝒙∙𝒙∙𝒙∙𝒙∙𝒙∙𝒚∙𝒚∙𝒚∙𝒚∙𝒚 𝟓 𝐲 =𝟐∙𝟑∙ 𝒙 𝟑 ∙ 𝒚 𝟐 𝟓𝒚 =𝟔 𝒙 𝟑 𝒚 𝟐 𝟓𝒚

𝟑 𝒚 Mrs. Rivas You Do It = 𝟏𝟓 𝒙 𝟓 𝒚 𝟑 𝟐𝟎𝒙 𝒚 𝟒 = 𝟏𝟓∙𝟐𝟎 𝒙 𝟓+𝟏 𝒚 𝟑+𝟒 ISCHS Section 6-2 Multiplying and Dividing Radical Expressions. You Do It Simplify a product. What is the simplest form of 15 𝑥 5 𝑦 3 ∙ 20𝑥 𝑦 4 ? = 𝟏𝟓 𝒙 𝟓 𝒚 𝟑 𝟐𝟎𝒙 𝒚 𝟒 = 𝟏𝟓∙𝟐𝟎 𝒙 𝟓+𝟏 𝒚 𝟑+𝟒 = 𝟐∙𝟐∙𝟑∙𝟓∙𝟓∙𝒙∙𝒙∙𝒙∙𝒙∙𝒙∙𝒙∙𝒚∙𝒚∙𝒚∙𝒚∙𝒚∙𝒚∙𝒚 𝟑 𝒚 =𝟐∙𝟓∙ 𝒙 𝟑 ∙ 𝒚 𝟑 𝟑𝒚 =𝟏𝟎 𝒙 𝟑 𝒚 𝟑 𝟑𝒚

Mrs. Rivas ISCHS Section 6-2 Multiplying and Dividing Radical Expressions. Objective: To divide radical expressions 𝒏 𝑎 𝒏 𝑏 = 𝒏 𝑎 𝑏

Dividing radical expressions. Mrs. Rivas ISCHS Section 6-2 Multiplying and Dividing Radical Expressions. Example # 4 Dividing radical expressions. What is the simplest form of each quotient. = 𝟏𝟖 𝒙 𝟓 𝟐 𝒙 𝟑 A 𝟏𝟖𝒙 𝟓 𝟐𝒙 𝟑 = 𝟑 𝟏𝟔𝟐 𝒚 𝟓 𝟑 𝒚 𝟐 B 𝟑 𝟏𝟔𝟐𝒚 𝟓 𝟑 𝟑𝒚 𝟐 = (𝟏𝟖÷𝟐)( 𝒙 𝟓−𝟑 ) = 𝟑 (𝟏𝟔𝟐÷𝟑)( 𝒚 𝟓−𝟐 ) = 𝟑 𝟑∙𝟑∙𝟑∙𝟐∙ (𝒚) 𝟑 = 𝟗 𝒙 𝟐 = 𝟑 𝟓𝟒 𝒚 𝟑 =𝟑𝒚 𝟑 𝟐 =𝟑 𝒙

Mrs. Rivas You Do It = 𝟓𝟎 𝒙 𝟔 𝟐 𝒙 𝟒 = (𝟓𝟎÷𝟐)( 𝒙 𝟔−𝟒 ) = 𝟐𝟓 𝒙 𝟐 =𝟓 𝒙 ISCHS Section 6-2 Multiplying and Dividing Radical Expressions. You Do It Dividing radical expressions. What is the simplest form of 50𝑥 6 2𝑥 4 ? = 𝟓𝟎 𝒙 𝟔 𝟐 𝒙 𝟒 = (𝟓𝟎÷𝟐)( 𝒙 𝟔−𝟒 ) = 𝟐𝟓 𝒙 𝟐 =𝟓 𝒙

Rationalizing the denominator. Mrs. Rivas ISCHS Section 6-2 Multiplying and Dividing Radical Expressions. Example # 5 Rationalizing the denominator. What is the simplest form of 3 5𝑥 2 12𝑦 2 𝑧 ? = 𝟑 𝟓 𝒙 𝟐 𝟑 𝟏𝟐 𝒚 𝟐 𝒛 = 𝟑 𝟓 𝒙 𝟐 𝟑 𝟐∙𝟐∙𝟑∙ 𝒚 𝟐 ∙𝒛 × 𝟑 𝟐∙ 𝟑 𝟐 ∙𝒚∙ 𝒛 𝟐 𝟑 𝟐∙ 𝟑 𝟐 ∙𝒚∙ 𝒛 𝟐 = 𝟑 (𝟓∙𝟐∙ 𝟑 𝟐 )∙ 𝒙 𝟐 ∙𝒚∙ 𝒛 𝟐 𝟑 (𝟐∙𝟐∙𝟐)(𝟑∙𝟑∙𝟑)( 𝒚 𝟐+𝟏 )( 𝒛 𝟏+𝟐 ) = 𝟑 𝟗𝟎 𝒙 𝟐 𝒚 𝒛 𝟐 𝟐∙𝟑∙𝒚∙𝒛 = 𝟑 𝟗𝟎 𝒙 𝟐 𝒚 𝒛 𝟐 𝟔𝒚𝒛

Mrs. Rivas You Do It × 𝟑 𝟓 𝟐 ∙𝒚 𝟑 𝟓 𝟐 ∙𝒚 = 𝟑 𝟕𝒙 𝟑 𝟓 𝒚 𝟐 ISCHS Section 6-2 Multiplying and Dividing Radical Expressions. You Do It Rationalizing the denominator. What is the simplest form of 3 7𝑥 5𝑦 2 ? = 𝟑 𝟕𝒙 𝟑 𝟓 𝒚 𝟐 × 𝟑 𝟓 𝟐 ∙𝒚 𝟑 𝟓 𝟐 ∙𝒚 = 𝟑 (𝟕∙𝟓∙𝟓)𝒙∙𝒚 𝟑 (𝟓∙𝟓∙𝟓) 𝒚 𝟐+𝟏 = 𝟑 𝟏𝟕𝟓𝒙𝒚 𝟓𝒚

Binomial Radical Expressions. Mrs. Rivas ISCHS Section 6-3 Binomial Radical Expressions. Objective: To add and subtract radical expressions Like radicals: are radical expressions that have the same index and radicand. 𝟐 +𝟑 𝟐 =𝟒 𝟐 𝟓𝒙𝒚 +𝟖 𝟓𝒙𝒚 =𝟗 𝟓𝒙𝒚 𝟑 𝟕 −𝟓 𝟑 𝟕 =−𝟒 𝟑 𝟕 𝟑 𝟗 𝒙 𝟐 𝒚 −𝟖 𝟑 𝟗 𝒙 𝟐 𝒚 =−𝟕 𝟑 𝟗 𝒙 𝟐 𝒚

𝑎 𝑛 𝑥 +𝑏 𝒏 𝑥 =(𝑎+𝑏) 𝒏 𝑥 𝑎 𝑛 𝑥 −𝑏 𝒏 𝑥 =(𝑎−𝑏) 𝒏 𝑥 Mrs. Rivas ISCHS Section 6-3 Binomial Radical Expressions. 𝑎 𝑛 𝑥 +𝑏 𝒏 𝑥 =(𝑎+𝑏) 𝒏 𝑥 𝑎 𝑛 𝑥 −𝑏 𝒏 𝑥 =(𝑎−𝑏) 𝒏 𝑥

Binomial Radical Expressions. Mrs. Rivas ISCHS Section 6-3 Binomial Radical Expressions. Example # 1 Simplifying before adding or subtracting. What is the simplest form of the expression? 12 + 75 − 3 12 + 75 − 3 = 2∙2∙3 + 3∙5∙5 − 3 =2 3 +5 3 − 3 =7 3 − 3 =𝟔 𝟑

Binomial Radical Expressions. Mrs. Rivas ISCHS Section 6-3 Binomial Radical Expressions. You Do It Simplifying before adding or subtracting. What is the simplest form of the expression? 3 250 + 3 54 − 3 16 3 250 + 3 54 − 3 16 = 3 2∙5∙5∙5 + 3 2∙3∙3∙3 − 3 2∙2∙2∙2 =5 3 2 +3 3 2 −2 3 2 =8 3 2 −2 3 2 =𝟔 𝟑 𝟐

𝒏 ∙ 𝒏 = 𝒏 𝟐 =𝒏 Binomial Radical Expressions. Square root Property Mrs. Rivas ISCHS Section 6-3 Binomial Radical Expressions. Objective: Multiply binomial radical expressions Square root Property 𝒏 ∙ 𝒏 = 𝒏 𝟐 =𝒏 Ex. 5 ∙ 5 = 5 2 =𝟓 Ex. 3𝑥𝑦 ∙ 3𝑥𝑦 = 3 2 𝑥 2 𝑦 2 =𝟑𝒙𝒚 Ex. 3 7 ∙ 7 =3 7 2 =3(7) =𝟐𝟏

𝒂 ( 𝒃 ∙ 𝒄 )= 𝒂𝒃 ∙ 𝒂𝒄 Binomial Radical Expressions. Distribute Roots Mrs. Rivas ISCHS Section 6-3 Binomial Radical Expressions. Objective: Multiply binomial radical expressions Distribute Roots 𝒂 ( 𝒃 ∙ 𝒄 )= 𝒂𝒃 ∙ 𝒂𝒄 Ex. 2 8 + 10 = 16 + 20 =4+ 2∙2∙5 =𝟒+𝟐 𝟓 Rational number Irrational number

Binomial Radical Expressions. Mrs. Rivas ISCHS Section 6-3 Binomial Radical Expressions. You Do It A 𝟑 𝟏𝟐 − 𝟐𝟒 B 𝟏𝟓 𝟑 +𝟐 𝟓 𝟑𝟔 − 𝟕𝟐 𝟒𝟓 + 𝟐 𝟕𝟓 𝟔− 𝟐∙𝟐∙𝟐∙𝟑∙𝟑 𝟑∙𝟑∙𝟓 +𝟐 𝟑∙𝟓∙𝟓 𝟔−𝟔 𝟐 𝟑 𝟓 +𝟐∙𝟓 𝟑 𝟑 𝟓 +𝟏𝟎 𝟑

Binomial Radical Expressions. Mrs. Rivas ISCHS Section 6-3 Binomial Radical Expressions. Example # 2 Multiplying binomial radical expressions. What is the product of each expression? A (𝟒+𝟐 𝟐 )(𝟓+𝟒 𝟐 ) (𝟒+𝟐 𝟐 )(𝟓+𝟒 𝟐 ) 𝟐𝟎 + 𝟏𝟔 𝟐 + 𝟏𝟎 𝟐 + 𝟖 𝟒 𝟐𝟎+𝟐𝟔 𝟐 +𝟖(𝟐) 𝟐𝟎+𝟐𝟔 𝟐 +𝟏𝟔 𝟑𝟔+𝟐𝟔 𝟐

Binomial Radical Expressions. Mrs. Rivas ISCHS Section 6-3 Binomial Radical Expressions. Example # 2 Multiplying binomial radical expressions. What is the product of each expression? B (𝟑− 𝟕 )(𝟓+ 𝟕 ) (𝟑− 𝟕 )(𝟓+ 𝟕 ) 𝟏𝟓 + 𝟑 𝟕 − 𝟓 𝟕 − 𝟒𝟗 𝟏𝟓−𝟐 𝟕 −𝟕 𝟖−𝟐 𝟕

Binomial Radical Expressions. Mrs. Rivas ISCHS Section 6-3 Binomial Radical Expressions. You Do It Simplifying before adding or subtracting. What is product (3+2 5 )(2+4 5 )? (𝟑+𝟐 𝟓 )(𝟐+𝟒 𝟓 ) 𝟔 + 𝟏𝟐 𝟓 + 𝟒 𝟓 + 𝟖 𝟐𝟓 𝟔+𝟏𝟔 𝟓 +𝟖(𝟓) 𝟔+𝟏𝟔 𝟓 +𝟒𝟎 𝟒𝟔+𝟏𝟔 𝟓

Binomial Radical Expressions. Mrs. Rivas ISCHS Section 6-3 Binomial Radical Expressions. Conjugates: are expressions, like 𝒂 + 𝒃 and 𝒂 − 𝒃 , that differ only in the signs of the second term. When 𝑎 and 𝑏 are rational numbers, the product of two radical conjugates in a rational number. Hint The difference of squares factoring is 𝒂 𝟐 − 𝒃 𝟐 = 𝒂+𝒃 𝒂−𝒃 .

Binomial Radical Expressions. Mrs. Rivas ISCHS Section 6-3 Binomial Radical Expressions. Example # 3 Multiplying Conjugates. What is product (5− 7 )(5+ 7 )? (𝟓 − 𝟕 )(𝟓+ 𝟕 ) 𝟐𝟓 + 𝟓 𝟕 − 𝟓 𝟕 − 𝟒𝟗 𝟐𝟓−𝟎 𝟕 −𝟕 𝟏𝟖

Binomial Radical Expressions. Mrs. Rivas ISCHS Section 6-3 Binomial Radical Expressions. You Do It Multiplying Conjugates. What is each product? A (𝟔− 𝟏𝟐 )(𝟔+ 𝟏𝟐 ) B (𝟑+ 𝟖 )(𝟑− 𝟖 ) (𝟔 − 𝟏𝟐 )(𝟔+ 𝟏𝟐 ) (𝟑+ 𝟖 )(𝟑− 𝟖 ) 𝟑𝟔 + 𝟔 𝟏𝟐 − 𝟔 𝟏𝟐 − 𝟏𝟒𝟒 𝟗 − 𝟑 𝟖 + 𝟑 𝟖 − 𝟔𝟒 𝟑𝟔−𝟎 𝟏𝟐 −𝟏𝟐 𝟗−𝟎 𝟖 −𝟖 𝟐𝟒 𝟏

Binomial Radical Expressions. Mrs. Rivas ISCHS Section 6-3 Binomial Radical Expressions. Example # 5 Rationalizing the denominator. How can you write the expression with a rationalized denominator? 𝟑 𝟐 𝟓 − 𝟐 × 𝟓 + 𝟐 𝟓 + 𝟐 Multiply the top and bottom by the conjugate of the denominator. = 𝟑 𝟐 𝟓 + 𝟐 𝟓 𝟐 − 𝟐 𝟐 = 𝟑 𝟏𝟎 +𝟑 𝟒 𝟓−𝟐 = 𝟑 𝟏𝟎 𝟑 + 𝟔 𝟑 = 𝟑 𝟏𝟎 +𝟑(𝟐) 𝟑 = 𝟏𝟎 +𝟐

Binomial Radical Expressions. Mrs. Rivas ISCHS Section 6-3 Binomial Radical Expressions. You Do It Rationalizing the denominator. How can you write the expression with a rationalized denominator? × 𝟑 + 𝟓 𝟑 + 𝟓 × 𝟑+ 𝟔 𝟑+ 𝟔 A 𝟐 𝟕 𝟑 − 𝟓 B 𝟒𝒙 𝟑− 𝟔 = 𝟒𝒙 𝟑+ 𝟔 𝟑 𝟐 − 𝟔 𝟐 = 𝟐 𝟕 𝟑 + 𝟓 𝟑 𝟐 − 𝟓 𝟐 = 𝟐 𝟐𝟏 +𝟐 𝟑𝟓 −𝟐 = 𝟏𝟐𝒙+𝟒𝒙 𝟔 𝟗−𝟔 = 𝟏𝟐𝒙 𝟑 + 𝟒𝒙 𝟔 𝟑 = 𝟐 𝟐𝟏 −𝟐 + 𝟐 𝟑𝟓 −𝟐 =− 𝟐𝟏 − 𝟑𝟓 =𝟒𝒙+ 𝟒𝒙 𝟔 𝟑

𝒂 𝒎 𝒏 = 𝒏 𝒂 𝒎 = 𝒏 𝒂 𝒎 Rational Exponents Mrs. Rivas ISCHS Section 6-4 Rational Exponents Objective: To use rational exponents. 𝒂 𝒎 𝒏 = 𝒏 𝒂 𝒎 = 𝒏 𝒂 𝒎

Rational Exponents Mrs. Rivas Example # 1 ISCHS Section 6-4 Rational Exponents Example # 1 Simplifying Expressions with rational exponents. What is the simplest form of each expression? 𝟐𝟏𝟔 𝟏 𝟑 A B 𝟕 𝟏 𝟐 ∙ 𝟕 𝟏 𝟐 𝟐𝟏𝟔 𝟏 𝟑 = 𝟑 𝟐𝟏𝟔 𝟕 𝟏 𝟐 ∙ 𝟕 𝟏 𝟐 = 𝟕 ∙ 𝟕 = 𝟑 𝟐∙𝟐∙𝟐∙𝟑∙𝟑∙𝟑 = 𝟒𝟗 =𝟔 =𝟕 Note: exponent of 1 2 is the same as Note: exponent of 1 3 is the same as 3

Rational Exponents Mrs. Rivas Example # 1 ISCHS Section 6-4 Rational Exponents Example # 1 Simplifying Expressions with rational exponents. What is the simplest form of each expression? 𝟓 𝟏 𝟒 ∙ 𝟏𝟐𝟓 𝟏 𝟒 C 𝟓 𝟏 𝟒 ∙ 𝟏𝟐𝟓 𝟏 𝟒 = 𝟒 𝟓 ∙ 𝟒 𝟏𝟐𝟓 = 𝟒 𝟓∙𝟓∙𝟓∙𝟓 =𝟓 Note: exponent of 1 4 is the same as 4

𝒂 𝒎 𝒏 = 𝒏 𝒂 𝒎 Rational Exponents = 𝟏 𝒙 𝟏 𝟑 = 𝟏 𝟑 𝒙 1) 𝒙 𝟐 𝟗 2) 𝒙 −𝟏 𝟑 Mrs. Rivas ISCHS Section 6-4 Rational Exponents Example # 2 Converting between exponential and radical form. 𝒂 𝒎 𝒏 = 𝒏 𝒂 𝒎 = 𝟏 𝒙 𝟏 𝟑 = 𝟏 𝟑 𝒙 1) 𝒙 𝟐 𝟗 2) 𝒙 −𝟏 𝟑 = 𝟗 𝒙 𝟐 = 𝟏 𝒚 𝟕 𝟐 = 𝟏 𝒚 𝟕 = 𝒚 −𝟕 𝟐 3) 𝒚 −𝟑.𝟓

𝒏 𝒂 𝒎 =𝒂 𝒎 𝒏 Rational Exponents 1) 𝟗 = 𝟗 𝟏 𝟐 2) 𝟑 𝒙 𝟐 = 𝒙 𝟐 𝟑 3) 𝒂 𝟑 Mrs. Rivas ISCHS Section 6-4 Rational Exponents Example # 3 Converting between exponential and radicals form. =𝒂 𝒎 𝒏 𝒏 𝒂 𝒎 1) 𝟗 = 𝟗 𝟏 𝟐 2) 𝟑 𝒙 𝟐 = 𝒙 𝟐 𝟑 3) 𝒂 𝟑 = 𝒂 𝟑 𝟐 4) 𝟑 𝒂 𝟐 = 𝒂 𝟐 𝟑

Rational Exponents 𝒂 𝒎 ∙ 𝒂 𝒏 = 𝒂 𝒎+𝒏 𝒂 𝒎 𝒏 = 𝒂 𝒎 ∙ 𝒏 𝒂𝒃 𝒎 = 𝒂 𝒎 ∙ 𝒃 𝒎 Mrs. Rivas ISCHS Section 6-4 Rational Exponents Objective: To use rational exponents. Properties Examples 𝟖 𝟏 𝟑 ∙ 𝟖 𝟐 𝟑 = 𝟖 𝟏 𝟑 + 𝟐 𝟑 = 𝟖 𝟏 =𝟖 𝒂 𝒎 ∙ 𝒂 𝒏 = 𝒂 𝒎+𝒏 𝟓 𝟏 𝟐 𝟒 = 𝟓 𝟏 𝟐 ∙ 𝟒 𝟏 = 𝟓 𝟐 =𝟐𝟓 𝒂 𝒎 𝒏 = 𝒂 𝒎 ∙ 𝒏 𝟒∙𝟓 𝟏 𝟐 = 𝟒 𝟏 𝟐 ∙ 𝟓 𝟏 𝟐 = 𝟐∙𝟓 𝟏 𝟐 𝒂𝒃 𝒎 = 𝒂 𝒎 ∙ 𝒃 𝒎

Rational Exponents 𝒂 −𝒎 = 𝟏 𝒂 𝒎 𝒂 𝒎 𝒂 𝒏 = 𝒂 𝒎 − 𝒏 𝒂 𝒃 𝒎 = 𝒂 𝒎 𝒃 𝒎 Mrs. Rivas ISCHS Section 6-4 Rational Exponents Objective: To use rational exponents. Properties Examples 𝒂 −𝒎 = 𝟏 𝒂 𝒎 𝟗 − 𝟏 𝟐 = 𝟏 𝟗 𝟏 𝟐 = 𝟏 𝟑 𝟕 𝟑 𝟐 𝟕 𝟏 𝟐 = 𝟕 𝟑 𝟐 − 𝟏 𝟐 = 𝟕 𝟏 =𝟕 𝒂 𝒎 𝒂 𝒏 = 𝒂 𝒎 − 𝒏 𝟓 𝟐𝟕 𝟏 𝟑 = 𝟓 𝟏 𝟑 𝟐𝟕 𝟏 𝟑 = 𝟓 𝟏 𝟑 𝟑 𝒂 𝒃 𝒎 = 𝒂 𝒎 𝒃 𝒎

Rational Exponents = 𝒙 𝟑 𝟒 𝒙 𝟐 𝟖 𝟒 𝒙 𝟑 𝟖 𝒙 𝟐 = 𝒙 𝟑 𝟒 − 𝟏 𝟒 Mrs. Rivas ISCHS Section 6-4 Rational Exponents Example # 4 Combining Radicals. What is 𝟒 𝒙 𝟑 𝟖 𝒙 𝟐 in simplest form? 𝒂 𝒎 𝒂 𝒏 = 𝒂 𝒎 − 𝒏 = 𝒙 𝟑 𝟒 𝒙 𝟐 𝟖 𝟒 𝒙 𝟑 𝟖 𝒙 𝟐 = 𝒙 𝟑 𝟒 − 𝟏 𝟒 = 𝒙 𝟏 𝟐 𝐨𝐫 𝒙

Rational Exponents 𝟔 𝒙 𝟓 𝟒 𝟐𝟕 Mrs. Rivas You Do It 𝒙 𝟑 𝟑 𝒙 𝟐 A B 𝟑 𝟒 𝟑 ISCHS Section 6-4 Rational Exponents You Do It Simplifying Expressions with rational exponents. What is each quotient or product in simplest form? 𝒙 𝟑 𝟑 𝒙 𝟐 A 𝒂 𝒎 𝒂 𝒏 = 𝒂 𝒎 − 𝒏 B 𝟑 𝟒 𝟑 𝒂 𝒎 ∙ 𝒂 𝒏 = 𝒂 𝒎+𝒏 𝟔 𝒙 𝟓 𝟒 𝟐𝟕

Rational Exponents Mrs. Rivas Example # 5 ISCHS Section 6-4 Rational Exponents Example # 5 Simplifying Numbers with Rational Exponents. What is each number in simplest form? −𝟑𝟐 𝟒 𝟓 A 𝟏𝟔 −𝟐.𝟓 B 𝟏𝟔 − 𝟓 𝟐 = 𝟏 𝟏𝟔 𝟓 𝟐 = 𝟏 𝟏𝟔 𝟓 −𝟑𝟐 𝟒 𝟓 = 𝟓 −𝟑𝟐 𝟒 = −𝟐 𝟒 = 𝟏 𝟒 𝟓 =𝟏𝟔 = 𝟏 𝟏𝟎𝟐𝟒

Rational Exponents 𝟏 𝟐 𝒙 𝟓 Mrs. Rivas You Do It ISCHS Section 6-4 Rational Exponents You Do It Writing expressions in simplest form. What is 𝟖 𝒙 𝟏𝟓 − 𝟏 𝟑 each expression in simplest form? 𝒂 −𝒎 = 𝟏 𝒂 𝒎 𝒂𝒃 𝒎 = 𝒂 𝒎 ∙ 𝒃 𝒎 𝟏 𝟐 𝒙 𝟓

Mrs. Rivas ISCHS Pg. 422-424 # 1-43 All