answer:# Solution: For the first problem: We start by breaking down the expression into parts and solving each part step by step: 1. (-2)^2 = 4 because when you square a negative number, it becomes positive. 2. (3-pi)^0 = 1 because any non-zero number raised to the power of 0 equals 1. 3. (frac{1}{3})^{-1} = 3 because the negative exponent means we take the reciprocal, and the reciprocal of frac{1}{3} is 3. Putting it all together: (-2)^2 - (3-pi)^0 + (frac{1}{3})^{-1} = 4 - 1 + 3 = 6. So, the final answer for the first part is boxed{6}. For the second problem: We will expand and simplify the expression step by step: 1. Expand a(a+2) to get a^2 + 2a. 2. Expand (a-1)(a+3) to get a^2 + 3a - a - 3 which simplifies to a^2 + 2a - 3. 3. Subtract the expanded forms: (a^2 + 2a) - (a^2 + 2a - 3). Simplifying the expression: a^2 + 2a - a^2 - 2a + 3 = 3. So, the final answer for the second part is boxed{3}.

answer:Given 5x + 3 equiv 1 pmod{18}, we first normalize the equation to isolate x: 5x equiv -2 pmod{18} Here, -2 is equivalent to 16 modulo 18, so: 5x equiv 16 pmod{18} The next step involves finding the multiplicative inverse of 5 modulo 18. We use the extended Euclidean algorithm, which confirms that 5 times 11 equiv 1 pmod{18}. Therefore, the inverse of 5 modulo 18 is 11. Multiply both sides of the equation by 11: x equiv 11 times 16 pmod{18} Calculate: 11 times 16 = 176 And reducing modulo 18, we find: 176 equiv 8 pmod{18} Thus, x equiv 8 pmod{18}. Now to find 3x + 8 pmod{18}: 3x + 8 equiv 3(8) + 8 pmod{18} Calculate: 24 + 8 = 32 And reducing modulo 18, we find: 32 equiv 14 pmod{18} Thus, 3x + 8 equiv boxed{14} pmod{18}.

answer:Let's break down the steps and calculate the total step by step: 1. Start with 1000. 2. Add an unknown number (let's call it X). 3. Add 1000 again. 4. Add 30. 5. Add 1000 again. 6. Add 40. 7. Add 1000 again. 8. Add 10. The total is said to be 4100. So the equation based on the steps is: 1000 + X + 1000 + 30 + 1000 + 40 + 1000 + 10 = 4100 Combine all the known numbers: 1000 + 1000 + 1000 + 1000 + 30 + 40 + 10 = 4100 4000 + 80 = 4100 Now subtract 4000 and 80 from both sides to find X: 4100 - 4000 - 80 = X 100 - 80 = X 20 = X The first number added to 1000 is boxed{20} .

answer:If the lightsaber toy is twice as expensive as all his other Star Wars toys combined, and his other toys cost 1000, then the lightsaber costs 2 * 1000 = 2000. After buying the lightsaber, the total amount spent on Star Wars toys would be the cost of the other toys plus the cost of the lightsaber, which is 1000 + 2000 = boxed{3000} .