Questions and Answers #1 Rational Exponents and Radicals

Questions and Answers Sheet 1 Rational Exponents and Radicals Question #1 Use a graphing calculator to determine the exponential regression equation to model the relationship between the decades since 1950 and the population. a. The calculator returns two values, a and b. Write these values below. Round a to the nearest whole number and b to the nearest thousandth, if necessary. a=?,b=?. b. The general form of an exponential function is 𝑦 = 𝑎𝑏 𝑥 . b. The general form of an exponential function is 𝑦 = 𝑎𝑏 𝑥 . Use this general form and the values of a and b from Part (a) to write an exponential function that models Nevada's population growth. c. Use a graphing calculator to graph the data points and the function from Part (b). Sketch the graph and the data points below. Is the exponential function a good approximation of the data? Answer: Step 1 a. Values of a and b a=182,753 and b=1.612 b. Function The exponential function that represents the population growth of Nevada 𝑓(𝑥) = 182,753 ⋅ (1.612) 𝑥 Step 2 c. Graph of the data blue - graph of the function orange - graph of the data points The exponential function is a good approximation of the data as shown on the graph since the graph of the function is closer to the graph of the data points. Question #2 Write the radical expression using exponents and each exponential expression using radicals. 3 √72 Answer: Step 1 3 The given expression is √72 Radical rule: 1 𝑛 √𝑎 = 𝑎 𝑛 Exponent rule: (𝑎𝑏 )𝑐 = 𝑎𝑏𝑐 Step 2 1 3 √72 = (72 )3 2 = 73 Question #3 Using exponents, write down the radical expression, then write them down as an exponential expression using radicals. 10−3/2 Answer: Step 1 3 Given exponential expression 102 By using 1 𝑎−𝑚 = 𝑚 𝑎 𝑎𝑚𝑛 = (𝑎𝑚 )𝑛 Step 2 Calculation: 3 1 102 = 1 = 3 102 1 3 (102 ) = = = 1 3 (√10) 1 2 (√10) ×√10 1 10√10 Question #4 Simplify expression and express your answer using rational indicators. Suppose all letters represent positive numbers. 1 5 √𝑥 3 Answer: Step 1 Concept used: If a - a real number, n - positive integer 𝑛 √𝑎𝑚 = 𝑎𝑚\𝑛 The above statement can be expressed as, 1 −𝑛 𝑛 = 𝑎 𝑎 Step 2 The property of n the root after combine the 1 𝑛 √𝑎𝑚 1 𝑎𝑛 𝑛 = 𝑎−𝑛 and √𝑎𝑚 = 𝑎𝑚/𝑛 is, = 𝑎 −𝑚/𝑛 Substitute 5 for n, 3 for m and x for a in the above equation. 1 1 5 3 = 1/5 3 √𝑥 (𝑥 ) 1/5 −3 = (𝑥 ) = 𝑥 −3/5 Thus, the solution of the expression 1 5 √𝑥 3 is 𝑥 −3/5 Question #5 Simplify the expression. Then express your answer using rational indicators. Let the letters stand for positive numbers. √𝑥𝑦 √16𝑥𝑦 4 Answer: Step 1 Formula: 𝑛 √𝑎𝑚 = 𝑎𝑚/𝑛 Law of exponent: 𝑎𝑚 𝑎𝑛 = 𝑎𝑚−𝑛 Step 2 Use the property of n the root, in the expression √𝑥𝑦 √16𝑥𝑦 4 (𝑥𝑦)1/2 √𝑥𝑦 √16𝑥𝑦 = (16)1/4(𝑥𝑦)1/4 4 (𝑥𝑦)1/2 = (24)1/4(𝑥𝑦)1/4 = = (𝑥𝑦)1/2 1 (2)4⋅4 (𝑥𝑦)1/4 (𝑥𝑦) 1/2 2(𝑥𝑦)1/4 Now, use the law of exponents, in the above expression. = = (𝑥𝑦) 1/2 2(𝑥𝑦)1/4 1 1 = (𝑥𝑦) 2−4 2 2−1 (𝑥𝑦) 4 2 1 = (𝑥𝑦) 4 2 Thus, the solution of the expression √𝑥𝑦 √16𝑥𝑦 4 is = (𝑥𝑦)1/4 2 Question #6 Simplify the expression. Express your answer using rational indicators if the letters represent positive numbers. √𝑎3 𝑏 4 √𝑎3 𝑏 2 Answer: Use the property of n the root, in the expression √𝑎3 𝑏 √𝑎3 𝑏 2 = 𝑎3/2 𝑏 1/2 = 4 𝑎3/4 𝑏 2/4 𝑎3/2 𝑏 1/2 𝑎3/4 𝑏 1/2 Now, use the law of exponents, in the above expression. 3 3 𝑎3/2 𝑏 1/2 1 1 ( ) ( ) = 𝑎 2−4 𝑏 2−2 𝑎3/4 𝑏 1/2 6−3 1−1 ( ) ( ) =𝑎 4 3 4 =𝑎 𝑏 𝑏 2 0 2 3 = 𝑎4 Thus, the solution of the expression √𝑎3 𝑏 4 √𝑎3 𝑏 2 is 𝑎3/4 Question #7 Solve the given expression using exponents. Write exponential expression using radicals. 1 √3 Answer: Solution: An exponential expression is of the form 𝑎𝑏 where a is called base and b is called the exponent. 1 also √𝑐 = (𝑐)2 1 1 √3 = 1 2 12 1 32 ∵1 =1 1 1 2 =( ) 3 𝑎𝑐 𝑎 𝑐 ∵ 𝑐=( ) 𝑏 𝑏 1 1 2 3 Therefore the exponential form is ( ) Question #8 Solve a radical expression using exponents. Exponential, solve by using radicals. 𝑎2/5 Answer: By using 𝑎𝑚𝑛 = (𝑎𝑚 )𝑛 2 1 Thus, 𝑎5 = (𝑎2 )5 5 = √𝑎2 5 2 Radical expression is √𝑎2 is 𝑎5 Question #9 The radical expression must be solved using exponents. Exponential, decide with radicals. 2−1.5 Answer: 2−1.5 = = = = 1 1 21.5 3 22 1 √ 23 1 2√2 Question #10 Rewrite the expression using rational exponents. 4 √𝑥 3 Answer: Step 1 4 Given radical expression: √𝑥 3 Step 2 To write Radicals as Exponents: 𝑛 𝑚 𝑛 𝑚 √𝑥 𝑚 = ( √𝑥 ) = 𝑥 𝑛 4 3 Therefore, √𝑥 3 = 𝑥 4 4 3 Hence, the radical expression √𝑥 3 is written in rational exponents as 𝑥 4