Questions and Answers #5 Rational Exponents and Radicals

Questions and Answers Sheet 5 Rational Exponents and Radicals Question #1 To calculate: 5 To evaluate the expression − √−1. Answer: Calculation: According to the basic rules of exponents, (𝑎𝑚 )𝑛 = 𝑎(𝑚)(𝑛) 5 This can be written as − √−1 = −(−1)1/5 5 − √−1 = −((−1)5 )1/5 i.e. 1𝑛 = 1 Applying the above rule, 5 − √−1 = −1(−1)(5)(1/5) 5 − √−1 = −(−1) 5 − √−1 = 1. Question #2 To calculate: 3 3 To simplify the expression √24 − √81. Answer: Calculation: When two similar irrational numbers are added or subtracted, the rational parts of the two numbers are added or subtracted. This can be written as: 3 3 3 √24 − √81 = √2 × 2 × 2 × 3 × 𝑥 − √3 × 3 × 3 × 3 × 𝑥 3 3 3 3 √24 − √81 = √23 × 3𝑥 − √33 × 3𝑥 3 3 3 3 3 3 √24 − √81 = √23 √3𝑥 − √33 √3𝑥 (Since the product rule for square root states that √𝑎𝑏 = √𝑎√𝑏) 3 3 3 3 √24 − √81 = 2 √3𝑥 = 3 √3𝑥 (Since cube and cube root are opposite operations, they cancel out) 3 3 3 √24 − √81 = (2 − 3) √3𝑥 3 3 3 √24 − √81 = (−1) √3𝑥 . Question #3 To calculate: To rationalize the numerator of the expression √𝑥−√5 . 𝑥 2 −25 Answer: Calculation: The numerator of a rational expression can be rationalized by removing the irrational term from it. To remove the irrational term, the numerator and denominator of the expression are multiplied by the conjugate of the numerator. Conjugate of an expression √𝑎 + √𝑏 is given as √𝑎 − √𝑏 and vice-versa. So the conjugate of the numerator is given as √𝑥 + √5. Multiplying the numerator and the denominator with the conjugate. √𝑥−√5 𝑥 2 −25 = √𝑥−√5 𝑥 2 −25 × √𝑥+√5 √𝑥+√5 (√𝑥−√5)(√𝑥+√5) √𝑥−√5 𝑥 2 −25 = √𝑥−√5 𝑥 2 −25 = (𝑥 2 −25)(√𝑥+√5) √𝑥−√5 = (𝑥 2 −25)(√𝑥+√5) 𝑥 2 −25 √𝑥−√5 (𝑥 2 −25)(√𝑥+√5) 2 (√𝑥) −(√5) 2 𝑖. 𝑒. (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏 2 𝑥−5 (Since square and square root are opposite operations, they cancel out) 𝑥−5 𝑥 2 −25 = (𝑥−5)(𝑥+5)( √𝑥−√5 𝑥 2 −25 = (𝑥+5)( √𝑥+√5) 𝑖. 𝑒. (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏 2 1 √𝑥+√5) Question #4 To calculate: To evaluate the expression (27𝑥 6 𝑦 3 )−2/3 . Answer: Calculation: Applying the fraction rule, −𝑎 𝑏 𝑎 =− 𝑏 2 The expression can be written as (27𝑥 6 𝑦 3 )−3 According to the basic rules of exponents, 𝑎−𝑚 = Applying the above rule, This can be written as (27𝑥 6 𝑦 3 )2/3 = 27 1 2 (27𝑥 6 𝑦 3 )3 1 2 (27𝑥 6 𝑦 3 )3 1 = 1 𝑎𝑚 1 2 (27𝑥 6 𝑦 3 )3 2/3 (𝑥 6 )2/3 (𝑦 3 )2/3 (According to exponent rule, (𝑎 ⋅ 𝑏)𝑛 = 𝑎𝑛 𝑏 𝑛 ) 1 272/3 (𝑥 6 )2/3 (𝑦 3 )2/3 1 = (33)2/3(𝑥 6)2/3(𝑦 3)2/3 1 2 (27𝑥 6 𝑦 3 )3 1 2 (27𝑥 6 𝑦 3 )3 = (3)3×2/3(𝑥)6×2/3(𝑦)3×2/3 = 1 9𝑥 4 𝑦 2 Question #5 To calculate: To evaluate the expression 15𝑥 3/2 3𝑥 2/4 . Answer: Calculation: Divide the numbers, 15 3 =5 The expression can be written as 5𝑥 3/2 𝑥 2/4 According to the basic rules of exponents, Applying the above rule, This can be written as 5𝑥 3/2 𝑥 2/4 5𝑥 3/2 = 5𝑥 12−4/8 = 5𝑥 8/8 𝑥 2/4 3/2 5𝑥 = 5𝑥. 𝑥 2/4 5𝑥 3/2 𝑥 2/4 = 5𝑥 3/2−2/4 𝑥𝑚 𝑥𝑛 = 𝑥 𝑚−𝑛 Question #6 To calculate: 3 To convert the radical expression √ √64𝑥 6 to rational exponent form and simplify. Answer: Calculation: 𝑛 1 According to exponential rules, √𝑎 = 𝑎 𝑛 Applying the above rule: 1 √ 3√64𝑥 6 = ( 3√64𝑥 6 )2 Applying the rule again: 1 1 √ 3√64𝑥 6 = ((64𝑥 6 )3 )2 According to the power rule, (𝑎𝑚 )𝑛 = 𝑎𝑚𝑛 Applying the above rule: 1 1 √ 3√64𝑥 6 = (64𝑥 6 )(3)(2) 1 √ 3√64𝑥 6 = (64𝑥 6 )(6) This can be written as 1 √ 3√64𝑥 6 = (26 𝑥 6 )(6) According to product rule, (𝑎𝑏)𝑛 = 𝑎𝑛 ⋅ 𝑏 𝑛 Applying the above rule: 1 1 √ 3√64𝑥 6 = (26 )(6) (𝑥 6 )(6) Applying the power rule: 1 1 √ 3√64𝑥 6 = 2(6)(6) 𝑥 (6)(6) √ 3√64𝑥 6 = 21 𝑥 1 √ 3√64𝑥 6 = 2𝑥 Question #7 To calculate: To simplify −6𝑥 2 − (−2𝑥 2 ). Answer: Calculation: −6𝑥 2 − (−2𝑥 2 ) This can be written as: −6𝑥 2 − (−2𝑥 2 ) = −6𝑥 2 + 2𝑥 2 −6𝑥 2 − (−2𝑥 2 ) = −4𝑥 2 . Question #8 To calculate: 3 To simplify the expression √−8𝑥 3 . Answer: Calculation: 𝑎 √𝑎 According to quotient property of square root √ = . 𝑏 √𝑏 And the product property of square root states √𝑎𝑏 = √𝑎 ⋅ √𝑏 3 3 This can be written as √−8𝑥 3 = √(−2) × (−2) × (−2) × 𝑥 × 𝑥 × 𝑥 3 3 3 3 √−8𝑥 3 = √(−2)3 × 𝑥 3 Applying the product rule, 3 √−8𝑥 3 = √(−2)3 × √𝑥 3 3 √−8𝑥 3 = (−2) × 𝑥 3 √−8𝑥 3 = −2𝑥 Question #9 Whether the given expression is true or false. 𝑚 Given: 𝑎 − 𝑛 = 1 𝑚 𝑎𝑛 Answer: Concept used: Reciprocal rule: 𝑎 −𝑥 = 1 𝑎𝑥 Calculation: 𝑚 On using the reciprocal rule, we will get 𝑎− 𝑛 = 1 𝑚 𝑎𝑛 Therefore, the given expression is true. Question #10 Whether the given expression is true or false. Given: √𝑥 2 = 𝑥 Answer: Concept used: 𝑛 𝑚 √𝑎 𝑚 = 𝑎 𝑛 Calculation: 2 On using the rule mentioned above, we will get √𝑥 2 = 𝑥 3 = 𝑥 Therefore, the given statement is true.