Simplify the following expressions: b. 8sqrt(2)sqrt(98)-3sqrt(2) and 2. a. (4^x)/(16) = ((8^x)/(8)) x sqrt(2)

b. Simplify without using a calculator 8sqrt(2)sqrt(98) - 3sqrt(2) Step 1: Simplify the radical in the denominator. sqrt(98) = sqrt(49 × 2) = sqrt(49) × sqrt(2) = 7sqrt(2) Step 2: Substitute the simplified radical back into the expression. 8sqrt(2)7sqrt(2) - 3sqrt(2) Step 3: Combine the terms in the denominator. 8sqrt(2)(7-3)sqrt(2) = 8sqrt(2)4sqrt(2) Step 4: Simplify the fraction. 8sqrt(2)4sqrt(2) = (8)/(4) = 2 The final answer is 2. 2. a. Solve for x in the equation below (4^x)/(16) = ((8^x)/(8)) × sqrt(2) Step 1: Express all terms with a base of 2. 4 = 2^2 16 = 2^4 8 = 2^3 sqrt(2) = 2^1/2 Step 2: Rewrite the equation using base 2. ((2^2)^x)/(2^4) = (((2^3)^x)/(2^3)) × 2^1/2 2^2x2^4 = 2^3x2^3 × 2^1/2 Step 3: Apply the exponent rules a^m/a^n = a^m-n and a^m × a^n = a^m+n. 2^2x-4 = 2^3x-3 × 2^1/2 2^2x-4 = 2^3x-3 + (1)/(2) 2^2x-4 = 2^3x - (6)/(2) + (1)/(2) 2^2x-4 = 2^3x - (5)/(2) Step 4: Equate the exponents and solve for x. 2x-4 = 3x - (5)/(2) Multiply the entire equation by 2 to eliminate the fraction: 2(2x-4) = 2(3x - (5)/(2)) 4x-8 = 6x - 5 4x - 6x = -5 + 8 -2x = 3 x = -(3)/(2) The final answer is -(3)/(2). b. The curve y = ax^2 + bx + c has a maximum value at (4, 6) and passes through the point (2, -2). Find the values of a, b and c. Step 1: Use the vertex form of a quadratic equation, y = a(x-h)^2 + k, where (h,k) is the vertex. Given the vertex is (4,6), substitute h=4 and k=6: y = a(x-4)^2 + 6 Step 2: Substitute the given point (2,-2) into the equation to find the value of a. -2 = a(2-4)^2 + 6 -2 = a(-2)^2 + 6 -2 = 4a + 6 -2 - 6 = 4a -8 = 4a a = -2 Step 3: Substitute the value of a back into the vertex form and expand to the standard form y = ax^2 + bx + c. y = -2(x-4)^2 + 6 y = -2(x^2 - 8x + 16) + 6 y = -2x^2 + 16x - 32 + 6 y = -2x^2 + 16x - 26 Step 4: Identify the values of a, b, and c by comparing with y = ax^2 + bx + c. a = -2 b = 16 c = -26 The final answer is a=-2, b=16, c=-26. What's next?