Express the following in index form: a) 81 b) 1/16. Classify the following as rational or irrational: a) sqrt(2) b) 0.25 c) pi

Here are the solutions to your mathematics questions. 3. Laws of Indices and Logarithms a) Express the following in index form: i) 81 Step 1: Find the prime factors of 81. 81 = 3 × 3 × 3 × 3 Step 2: Write in index form. 81 = 3^4 The index form is 3^4. ii) (1)/(16) Step 1: Express 16 in index form. 16 = 2 × 2 × 2 × 2 = 2^4 Step 2: Use the rule a^-n = (1)/(a^n). (1)/(16) = (1)/(2^4) = 2^-4 The index form is 2^-4. 1. Classify the following as rational or irrational: a) sqrt(2) Step 1: A rational number can be expressed as a fraction (p)/(q) where p and q are integers and q ≠ 0. An irrational number cannot. Step 2: The square root of 2 is a non-terminating, non-repeating decimal. It cannot be expressed as a simple fraction. Therefore, sqrt(2) is irrational. b) 0.25 Step 1: Express the decimal as a fraction. 0.25 = (25)/(100) Step 2: Simplify the fraction. (25)/(100) = (1)/(4) Since 0.25 can be expressed as the fraction (1)/(4), it is a rational number. Therefore, 0.25 is rational. c) Step 1: Recall the definition of . is the ratio of a circle's circumference to its diameter. Its decimal representation is non-terminating and non-repeating. Step 2: Since cannot be expressed as a simple fraction of two integers, it is an irrational number. Therefore, is irrational. d) (7)/(4) Step 1: A rational number can be expressed as a fraction (p)/(q) where p and q are integers and q ≠ 0. Step 2: The number (7)/(4) is already in the form of a fraction of two integers. Therefore, (7)/(4) is rational. 2. Find the reciprocal of: a) 5 Step 1: The reciprocal of a number x is (1)/(x). Step 2: For x=5, the reciprocal is (1)/(5). The reciprocal of 5 is (1)/(5). b) (2)/(3) Step 1: The reciprocal of a fraction (a)/(b) is (b)/(a). Step 2: For (2)/(3), the reciprocal is (3)/(2). The reciprocal of (2)/(3) is (3)/(2). 3 done, 2 left today. You're making progress.