Rewrite the equation using exponent rules.

Here's the solution to your problems: 6. Solve for x in 2^2x - 18 × 2^x = 40 Step 1: Rewrite the equation using exponent rules. We know that 2^2x = (2^x)^2. So the equation becomes: (2^x)^2 - 18 × 2^x = 40 Step 2: Introduce a substitution to form a quadratic equation. Let y = 2^x. Substitute y into the equation: y^2 - 18y = 40 Step 3: Rearrange the equation into standard quadratic form and solve for y. y^2 - 18y - 40 = 0 Factor the quadratic equation: (y - 20)(y + 2) = 0 This gives two possible values for y: y - 20 = 0 y = 20 y + 2 = 0 y = -2 Step 4: Substitute back 2^x for y and solve for x. Case 1: y = 20 2^x = 20 Take the logarithm of both sides (e.g., base 10 or natural log): (2^x) = (20) x (2) = (20) x = ((20))/((2)) Using a calculator: x ≈ (1.30103)/(0.30103) ≈ 4.3219 Case 2: y = -2 2^x = -2 An exponential function with a positive base, like 2^x, is always positive for real values of x. Therefore, there is no real solution for x in this case. The only real solution for x is: x = ((20))/((2)) ≈ 4.3219 7. Given that Sin(x - 30)^0 - Cos(4x)^0 = 0. Find the tan(2x+30)^0 Step 1: Rearrange the given equation. Sin(x - 30)^0 = Cos(4x)^0 Step 2: Use the trigonometric identity Sin() = Cos(90^0 - ). This means that the angles must be complementary (or differ by multiples of 360^0, but for basic problems, we assume the simplest case). So, we can equate the angles: (x - 30)^0 + (4x)^0 = 90^0 Step 3: Solve for x. x - 30 + 4x = 90 5x - 30 = 90 5x = 90 + 30 5x = 120 x = (120)/(5) x = 24 Step 4: Substitute the value of x into the expression tan(2x+30)^0. First, calculate the angle: 2x + 30 = 2(24) + 30 2x + 30 = 48 + 30 2x + 30 = 78 Now, find the tangent of this angle: tan(78)^0 Using a calculator: tan(78)^0 ≈ 4.7046 The value is: tan(2x+30)^0 ≈ 4.7046 3 done, 2 left today. You're making progress.