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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Step 1: Simplify 10^5 × 10^4. Using the rule a^m × a^n = a^m+n: 10^5 × 10^4 = 10^5+4 = 10^9 The simplified expression is 10^9. Step 2: Simplify a^3 × a^9. Using the rule a^m × a^n = a^m+n: a^3 × a^9 = a^3+9 = a^12 The simplified expression is a^12. Step 3: Simplify 5y × 4y^4. Multiply the coefficients and the variables separately: 5y × 4y^4 = (5 × 4) × (y^1 × y^4) = 20 × y^1+4 = 20y^5 The simplified expression is 20y^5. Step 4: Simplify 2^2 × 2^4. Using the rule a^m × a^n = a^m+n: 2^2 × 2^4 = 2^2+4 = 2^6 Calculate the value: 2^6 = 2 × 2 × 2 × 2 × 2 × 2 = 64 The simplified expression is 64. Step 5: Simplify m^8 ÷ m^5. Using the rule a^m ÷ a^n = a^m-n: m^8 ÷ m^5 = m^8-5 = m^3 The simplified expression is m^3. Step 6: Simplify c^7 ÷ c. Using the rule a^m ÷ a^n = a^m-n (where c = c^1): c^7 ÷ c^1 = c^7-1 = c^6 The simplified expression is c^6. Step 7: Simplify (24x^6)/(8x^4). Divide the coefficients and the variables separately: (24x^6)/(8x^4) = (24)/(8) × (x^6)/(x^4) = 3 × x^6-4 = 3x^2 The simplified expression is 3x^2. Step 8: Simplify (9 × 10^9)/(3 × 10^3). Divide the numerical parts and the powers of 10 separately: (9 × 10^9)/(3 × 10^3) = (9)/(3) × (10^9)/(10^3) = 3 × 10^9-3 = 3 × 10^6 The simplified expression is 3 × 10^6. Step 9: Simplify 2^0. Using the rule a^0 = 1 for any non-zero a: 2^0 = 1 The simplified expression is 1. Step 10: Simplify 6 × z^0. Using the rule a^0 = 1 for any non-zero a: 6 × z^0 = 6 × 1 = 6 The simplified expression is 6. Step 11: Simplify 4^-3. Using the rule a^-n = (1)/(a^n): 4^-3 = (1)/(4^3) Calculate the value: (1)/(4^3) = (1)/(4 × 4 × 4) = (1)/(64) The simplified expression is (1)/(64). Step 12: Simplify 3x^-3. Using the rule a^-n = (1)/(a^n): 3x^-3 = 3 × (1)/(x^3) = (3)/(x^3) The simplified expression is (3)/(x^3). Step 13: Simplify (1-4)^-2. First, simplify the expression inside the parenthesis: (1-4)^-2 = (-3)^-2 Using the rule a^-n = (1)/(a^n): (-3)^-2 = (1)/((-3)^2) Calculate the value: (1)/((-3)^2) = (1)/((-3) × (-3)) = (1)/(9) The simplified expression is (1)/(9). Step 14: Simplify (2-3)^-1. First, simplify the expression inside the parenthesis: (2-3)^-1 = (-1)^-1 Using the rule a^-n = (1)/(a^n): (-1)^-1 = (1)/((-1)^1) = (1)/(-1) = -1 The simplified expression is -1. Step 15: Simplify x^3 ÷ x^-5. Using the rule a^m ÷ a^n = a^m-n: x^3 ÷ x^-5 = x^3 - (-5) = x^3+5 = x^8 The simplified expression is x^8. Step 16: Simplify a^-9 ÷ b^0. Using the rule a^0 = 1 for any non-zero a: a^-9 ÷ b^0 = a^-9 ÷ 1 = a^-9 Using the rule a^-n = (1)/(a^n): a^-9 = (1)/(a^9) The simplified expression is (1)/(a^9). Step 17: Simplify (3x)^-3. Using the rule (ab)^n = a^n b^n: (3x)^-3 = 3^-3 x^-3 Using the rule a^-n = (1)/(a^n): 3^-3 x^-3 = (1)/(3^3) × (1)/(x^3) = (1)/(27) × (1)/(x^3) = (1)/(27x^3) The simplified expression is (1)/(27x^3). Step 18: Simplify 9a^-5 × 4a^6. Multiply the coefficients and the variables separately: 9a^-5 × 4a^6 = (9 × 4) × (a^-5 × a^6) = 36 × a^-5+6 = 36a^1 = 36a The simplified expression is 36a. Step 19: Simplify 5x^2 × 4x^0 × 2x^-6. Multiply the coefficients: 5 × 4 × 2 = 40 Multiply the variables using a^m × a^n = a^m+n: x^2 × x^0 × x^-6 = x^2+0+(-6) = x^2-6 = x^-4 Combine the results: 40x^-4 Using the rule a^-n = (1)/(a^n): 40x^-4 = (40)/(x^4) The simplified expression is (40)/(x^4). Step 20: Simplify 15 × 10^4 ÷ (3 × 10^-2). Rewrite the expression as a fraction: (15 × 10^4)/(3 × 10^-2) Separate the numerical parts and the powers of 10: = (15)/(3) × (10^4)/(10^-2) Simplify the numerical part: = 5 × (10^4)/(10^-2) Simplify the powers of 10 using a^m ÷ a^n = a^m-n: = 5 × 10^4 - (-2) = 5 × 10^4+2 = 5 × 10^6 The simplified expression is 5 × 10^6. That's 2 down. 3 left today — send the next one.