Questions and Answers #2 Rational Exponents and Radicals

Questions and Answers Sheet 2 Rational Exponents and Radicals Question #1 3 The simplified value of the expression √5𝑥 5 𝑦 3 √625𝑥 2 𝑦 4 Answer: Step 1 Formula used: If b is a real number and m and n are integers, then 𝑏𝑚 𝑏𝑛 𝑎 𝑚 = 𝑏 𝑚−𝑛 and ( ) = 𝑏 𝑛 𝑎𝑚 𝑏𝑚 𝑎𝑛 For an even integer n, √ 𝑛 = |𝑎| , and if n is odd, then √𝑎𝑛 = 𝑎 Step 2 Simplify the radicals by the properties of exponents. 3 √5𝑥 5 𝑦 3 =√ 3 √625𝑥 2 𝑦 4 𝑥 5−2 3 =√ 125𝑦 4−1 𝑥3 3 =√ 5𝑥 5 𝑦 625𝑥 2 𝑦 4 125𝑦 3 Forther solve and get, 3 √5𝑥 5 𝑦 3 =√ 3 √625𝑥 2 𝑦 4 3 𝑥 = √( ) 𝑥3 53 𝑦 3 3 5𝑦 = 𝑥 5𝑦 3 Therefore, the simplified value of √5𝑥 5 𝑦 3 √625𝑥 2 𝑦 4 is 𝑥 5𝑦 Question #2 𝑥 To calculate: √ 8 The simplified value of the radical expression. Answer: Step 1 Formula used: The relation between radical and rational exponent notation is expressed as, 𝑛 √𝑎 = 𝑎1/𝑛 Here, a is the radicand, and n is the index of the radical. 𝑚 𝑛 𝑛 ( √𝑎 ) = √ 𝑎 𝑚 = 𝑎𝑚/𝑛 The product rule for radicals, 𝑛 𝑛 𝑛 √𝑎𝑏 − √𝑎 ⋅ √𝑏 The quotient rule for radicals, 𝑛 𝑎 √𝑏 = 𝑛 √𝑎 𝑛 √𝑏 Step 2 Consider the given expression, 𝑥 √8 Use the relation between radical and rational exponent notation to simplify the expression, 𝑥 𝑥 1/2 √ =( ) 8 8 =( 𝑥 1/2 ) 4⋅2 𝑥 1/2 = (4)1/2 ⋅(2)1/2 𝑥 1/2 = (22)1/2 ⋅(2)1/2 Further simplify by using the product rule and quotient rule for radicals, 𝑥 √𝑥 √8 = 2√2 = = = √𝑥 × √2 2√2 √2 √𝑥⋅√2 2⋅√2⋅√2 √2𝑥 4 Hence, the simplified value of the provided expression is √2𝑥 4 Question #3 3 3𝑥 The simplified value of the expression √ 25 Answer: Step 1 Formula used: 𝑚 𝑛 𝑛 ( √𝑎 ) = √ 𝑎 𝑚 = 𝑎𝑚/𝑛 The product rule for radicals, 𝑛 𝑛 𝑛 √𝑎𝑏 = √𝑎 ⋅ √𝑏 The quotient rule for radicals, 𝑛 𝑛 𝑎 √𝑏 = √𝑎 𝑛 √𝑏 Step 2 Use the product rule for radicals to simplify the expression. 3 3𝑥 √ 3 3𝑥 =√ 25 3 3𝑥 52 =√ × 5⋅5 √5 √5 3 (3𝑥)(5) = √ (52)(5) Use the relation between radical and rational exponent notation to get the final simplified value. 3 3𝑥 √ 25 3 (15𝑥) = √ (53) 3 = √15𝑥 5 3 Hence, the simplified value of the provided expression is √15𝑥 5 Question #4 To calculate: The single radical of the expression √𝑥 √𝑥 √𝑥 Answer: Consider the expression √𝑥 √𝑥 √𝑥 1/2 1/2 √𝑥 √𝑥 √𝑥 = (𝑥 (𝑥(𝑥 1/2 )) ) Use the product property and the expression becomes, 1/2 1/2 (𝑥 (𝑥(𝑥 1/2 )) ) 1/2 1/2 = (𝑥(𝑥 1+1/2 ) ) 1/2 1/2 = (𝑥(𝑥 3/2 ) ) 3/4 1/2 (𝑥 ⋅ 𝑥 ) Add the powers of the same bases. 1/2 1/2 (𝑥 ⋅ 𝑥 3/4 ) = (𝑥 1+3/4 ) 1/2 = (𝑥 7/4 ) = 𝑥 7/8 The obtained expression with rational exponents can be rewritten into radicals as; 𝑥 7/8 = (𝑥 7 )1/8 8 = √𝑥 7 8 Therefore, the value of the expression is √𝑥 7 Question #5 3𝑦 To calculate: √ 20 The single radical of the expression. Answer: Formula: 𝑛 √𝑎 = 𝑎1/𝑛 𝑚 𝑛 𝑛 ( √𝑎 ) = √ 𝑎 𝑚 = 𝑎𝑚/𝑛 The product rule for radicals is: 𝑛 𝑛 𝑛 a) √𝑎𝑏 = √𝑎 ⋅ √𝑏 The quotient rule for radicals is: 𝑛 𝑛 𝑎 b) √ = 𝑏 √𝑎 𝑛 √𝑏 Calculation: a) √ =√ 3𝑦 20 3𝑦 1 2 b) √ = 3𝑦 5 3𝑦 20 1 3𝑦 2 5 = √ 1 √3⋅√5⋅√𝑦 2 √5⋅√5 1 √15⋅√𝑦 2 3𝑦 5⋅4 5⋅22 = √ = =√ 5 × √5 √5 = √15𝑦 10 Hence, the simplified value of the provided expression is √15𝑦 10 Question #6 To determine: The single radical of the expression 3 3 √𝑦 √𝑦 3√𝑦 Answer: Step 1 Formula used: Product property of radicals: If a and b are real numbers and n>1 is an integer, the product property is true provided that the radicals are real numbers. 𝑛 √𝑎𝑚 = 𝑎𝑚/𝑛 If m and n are integers and n>1 is an integer, then 𝑛 √𝑎𝑚 = 𝑎𝑚/𝑛 Step 2 3 3 Consider the expression √𝑦 √𝑦 3√𝑦 Rewrite the provided expression as rational exponents. 3 1/3 1/3 3 √𝑦 √𝑦 3√𝑦 = (𝑦 (𝑦(𝑦1/3 )) ) Use the product property and work from inside the parentheses and the expression becomes, 1/3 1/3 (𝑦 (𝑦(𝑦 1/3 )) ) 1/3 1/3 = (𝑦(𝑦1+1/3 ) ) 1/3 1/3 = (𝑦(𝑦 4/3 ) ) 4/9 1/3 = (𝑦 ⋅ 𝑦 ) Add the powers of the same bases. 1/3 (𝑦 ⋅ 𝑦 4/9 ) 1/3 = (𝑦1+4/9 ) 1/3 = (𝑦13/9 ) = 𝑦13/27 The obtained expression with rational exponents can be rewritten into radicals as; 𝑦13/27 = (𝑦13 )1/27 27 = √𝑦13 Therefore, the value of the expression is 27 √𝑦13 Question #7 3 1 The simplified value of the radical expression √ 2 Answer: Step 1 Formula: 𝑚 𝑛 𝑛 ( √𝑎 ) = √ 𝑎 𝑚 𝑛 𝑛 𝑛 a) √𝑎𝑏 = √𝑎 ⋅ √𝑏 𝑛 𝑛 𝑎 b) √ = 𝑏 √𝑎 𝑛 √𝑏 Step 2 3 1 3 1 2 2 3 √4 a) √ = √ × 3 √4 3 4 =√ 8 3 √4 = 3 √8 3 = √4 3 √23 3 3 1 √4 b) √ = (23)1/3 2 3 = √4 2 Question #8 To calculate: The single radical of the expression 3 √5 ⋅ √4 Answer: Step 1 The formula: 𝑛 𝑛 𝑛 a) √𝑎𝑏 = √𝑎 ⋅ √𝑏 𝑛 b) √𝑎 = 𝑎1/𝑛 Step 2 1 3 1 2 a) √5 ⋅ √4 = (5)2×3 ⋅ (4)3×2 3 3 2 = (5)6 ⋅ (4)6 6 6 = √53 ⋅ √42 3 6 b) √5 ⋅ √4 = √(5)3 (4)2 6 = √125 ⋅ 16 6 = √2000 Question #9 To calculate: The simplified value of the radical expression 5 ÷ √𝑥 Answer: Consider the given expression, 5 ÷ √𝑥 Use the product rule for radicals to simplify the expression, 5 5 ÷ √𝑥 = √𝑥 = 5 √𝑥 × √𝑥 √𝑥 = 5√𝑥 2 (√𝑥) Use the relation between radical and rational exponent notation to simplify the expression, 5 ÷ √𝑥 = 5√𝑥 2 (𝑥 1/2 ) 5√𝑥 = (𝑥)2(1/2) = 5√𝑥 𝑥 Hence, the simplified value of the provided expression is Question #10 1 To write the expression 25𝑥 2 in radical form. Answer: Given: 1 25𝑥 2 Concept used: 1 𝑛 𝑥 𝑛 = √𝑥 Calculation: 1 2 25𝑥 2 = 25 √𝑥 5√𝑥 𝑥