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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You're on a roll — let's solve these math problems! Question 2.1 To determine the value of the expression, we will use the exponent rule a^m+n = a^m · a^n and factor out common terms. Step 1: Factor out the lowest power of 2 from the numerator and the denominator. 2^2025 - 2^20232^2021 - 2^2019 = 2^2023(2^2025-2023 - 1)2^2019(2^2021-2019 - 1) = 2^2023(2^2 - 1)2^2019(2^2 - 1) Step 2: Simplify the terms in the parentheses and cancel common factors. = 2^2023(4 - 1)2^2019(4 - 1) = 2^2023(3)2^2019(3) = 2^20232^2019 Step 3: Use the exponent rule (a^m)/(a^n) = a^m-n to simplify further. = 2^2023 - 2019 = 2^4 = 16 The value of the expression is 16. Question 2.2.1 To simplify the expression, we will use exponent rules and factor out common terms. Step 1: Rewrite the terms in the numerator using a^m+n = a^m · a^n and factor out 3^x+1. 3^x+4 - 6 · 3^x+17 · 3^x+2 = 3^x+1 · 3^3 - 6 · 3^x+17 · 3^x+2 = 3^x+1(3^3 - 6)7 · 3^x+2 Step 2: Simplify the term in the parenthesis and rewrite 3^x+2 as 3^x+1 · 3^1. = 3^x+1(27 - 6)7 · 3^x+1 · 3^1 = 3^x+1(21)7 · 3^x+1 · 3 Step 3: Cancel out the common term 3^x+1 and simplify the numerical values. = (21)/(7 · 3) = (21)/(21) = 1 The simplified expression is 1. Question 2.2.2 To simplify the expression, we will use the property sqrt(A) × sqrt(B) = sqrt(AB) and the difference of squares formula (a+b)(a-b) = a^2 - b^2. Step 1: Combine the two square roots into a single square root. sqrt(x+4x-4) × sqrt(x-4x-4) = sqrt((x+4x-4))(x-sqrt(4x-4)) Step 2: Apply the difference of squares formula inside the square root. Let a=x and b=sqrt(4x-4). = sqrt(x^2 - (4x-4))^2 = sqrt(x^2 - (4x-4)) = sqrt(x^2 - 4x + 4) Step 3: Recognize the expression inside the square root as a perfect square trinomial. = sqrt((x-2)^2) = |x-2| The simplified expression is |x-2|. Question 2.3.1 To solve for x, we need to express both sides of the equation with the same base. Step 1: Rewrite 125 as a power of 5. 5^sqrt(x) = 5^3 Step 2: Since the bases are the same, equate the exponents. sqrt(x) = 3 Step 3: Square both sides to solve for x. (sqrt(x))^2 = 3^2 x = 9 The solution is x=9. Question 2.3.2 To solve for x, we will transform the equation into a quadratic form. Step 1: Rewrite 5^-x as (1)/(5^x). 5^x + 15 · (1)/(5^x) = 2 Step 2: Let y = 5^x. Substitute y into the equation. y + (15)/(y) = 2 Step 3: Multiply the entire equation by y to eliminate the denominator. y^2 + 15 = 2y Step 4: Rearrange the equation into standard quadratic form ay^2 + by + c = 0. y^2 - 2y + 15 = 0 Step 5: Calculate the discriminant = b^2 - 4ac to determine the nature of the roots. = (-2)^2 - 4(1)(15) = 4 - 60 = -56 Since the discriminant < 0, there are no real solutions for y. As y = 5^x must be a positive real number, there are no real solutions for x. The equation has no real solutions. Send me the next one 📸