This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the questions. Express the following in index form: (2)/(3) × (2)/(3) × (2)/(3) When a number is multiplied by itself multiple times, it can be written in index form. (2)/(3) × (2)/(3) × (2)/(3) = ((2)/(3))^3 The answer is ((2)/(3))^3. 10. Find the value of X: 5^X = 25 Step 1: Express 25 as a power of 5. 25 = 5^2 Step 2: Equate the powers. 5^X = 5^2 Since the bases are the same, the exponents must be equal. X = 2 11. Simplify: i) 5^3 × 5^4* Using the rule a^m × a^n = a^m+n: 5^3 × 5^4 = 5^3+4 = 5^7 ii) y^4 × y^4* Using the rule a^m × a^n = a^m+n: y^4 × y^4 = y^4+4 = y^8 12. Simplify and give your answer in index form: i) (4^2)^3* Using the rule (a^m)^n = a^m × n: (4^2)^3 = 4^2 × 3 = 4^6 ii) (y^4)^2* Using the rule (a^m)^n = a^m × n: (y^4)^2 = y^4 × 2 = y^8 13. Determine the value of: i) 3^0* Any non-zero number raised to the power of 0 is 1. 3^0 = 1 ii) 4^0 + 8^0* Using the rule a^0 = 1: 4^0 + 8^0 = 1 + 1 = 2 iii) 2^0 + 3^0* Using the rule a^0 = 1: 2^0 + 3^0 = 1 + 1 = 2 14. Simplify: i) 64^1/2* The power of (1)/(2) means taking the square root. 64^1/2 = sqrt(64) = 8 ii) 125^1/3* The power of (1)/(3) means taking the cube root. 125^1/3 = [3]125 = 5 That's 2 down. 3 left today — send the next one.