Rewrite 8 as 23 and apply the exponent rule a^m * a^n = a^m+n in the numerator.

Here are the solutions to the questions. 2.1 Simplify fully, without using a calculator: 2^n+1 - 8 · 2^n-32^n-2 Step 1: Rewrite 8 as 2^3 and apply the exponent rule a^m · a^n = a^m+n in the numerator. 2^n+1 - 2^3 · 2^n-32^n-2 = 2^n+1 - 2^3+(n-3)2^n-2 = 2^n+1 - 2^n2^n-2 Step 2: Factor out the common term 2^n from the numerator. (2^n(2^1 - 1))/(2^n-2) = (2^n(2 - 1))/(2^n-2) = (2^n · 1)/(2^n-2) = (2^n)/(2^n-2) Step 3: Apply the exponent rule (a^m)/(a^n) = a^m-n. 2^n - (n-2) = 2^n - n + 2 = 2^2 Step 4: Calculate the final value. 2^2 = 4 The simplified expression is 4. 2.2 Solve for x: 2.2.1 [x]27 = 2187 Step 1: Rewrite the radical as a fractional exponent and express both sides with base 3. 27^1/x = 2187 Since 27 = 3^3 and 2187 = 3^7: (3^3)^1/x = 3^7 Step 2: Apply the exponent rule (a^m)^n = a^mn. 3^3/x = 3^7 Step 3: Equate the exponents. (3)/(x) = 7 Step 4: Solve for x. 3 = 7x x = (3)/(7) The solution is x = (3)/(7). 2.2.2 4^x - 16 = 6 · 2^x Step 1: Rewrite 4^x as (2^2)^x = (2^x)^2. (2^x)^2 - 16 = 6 · 2^x Step 2: Let y = 2^x to form a quadratic equation. y^2 - 16 = 6y Rearrange into standard quadratic form: y^2 - 6y - 16 = 0 Step 3: Factor the quadratic equation. (y - 8)(y + 2) = 0 Step 4: Solve for y. y - 8 = 0 y = 8 y + 2 = 0 y = -2 Step 5: Substitute back 2^x for y and solve for x. For y = 8: 2^x = 8 2^x = 2^3 x = 3 For y = -2: 2^x = -2 There is no real solution for x since 2^x must always be positive. The solution is x = 3. 2.3 Given that x = sqrt(3) - 2, simplify (x^2+1)/(x^2-5) without using a calculator. (Give your answer in simplest surd form.) Step 1: Calculate x^2. x^2 = (sqrt(3) - 2)^2 x^2 = (sqrt(3))^2 - 2(sqrt(3))(2) + (-2)^2 x^2 = 3 - 4sqrt(3) + 4 x^2 = 7 - 4sqrt(3) Step 2: Substitute x^2 into the numerator x^2+1. x^2+1 = (7 - 4sqrt(3)) + 1 x^2+1 = 8 - 4sqrt(3) Step 3: Substitute x^2 into the denominator x^2-5. x^2-5 = (7 - 4sqrt(3)) - 5 x^2-5 = 2 - 4sqrt(3) Step 4: Form the fraction and simplify by factoring out common terms. (x^2+1)/(x^2-5) = 8 - 4sqrt(3)2 - 4sqrt(3) = 4(2 - sqrt(3))2(1 - 2sqrt(3)) = 2(2 - sqrt(3))1 - 2sqrt(3) Step 5: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of 1 - 2sqrt(3), which is 1 + 2sqrt(3). 2(2 - sqrt(3))1 - 2sqrt(3) × 1 + 2sqrt(3)1 + 2sqrt(3) Step 6: Expand the numerator. 2(2 - sqrt(3))(1 + 2sqrt(3)) = 2(2 · 1 + 2 · 2sqrt(3) - sqrt(3) · 1 - sqrt(3) · 2sqrt(3)) = 2(2 + 4sqrt(3) - sqrt(3) - 2 · 3) = 2(2 + 3sqrt(3) - 6) = 2(-4 + 3sqrt(3)) = -8 + 6sqrt(3) Step 7: Expand the denominator using the difference of squares formula (a-b)(a+b) = a^2 - b^2. (1 - 2sqrt(3))(1 + 2sqrt(3)) = 1^2 - (2sqrt(3))^2 = 1 - (4 · 3) = 1 - 12 = -11 Step 8: Combine the simplified numerator and denominator. -8 + 6sqrt(3)-11 = 8 - 6sqrt(3)11 The simplified expression is 8 - 6sqrt(3)11. Send me the next one 📸