Simplify the expression (4x4y^-2z16x2y^-1z^-2)^(1)/(2).

You're on a roll — Step 1: Simplify the expression (4x^4y^-2z16x^2y^-1z^-2)^(1)/(2). First, simplify the fraction inside the parentheses: 4x^4y^-2z16x^2y^-1z^-2 Simplify the coefficients: (4)/(16) = (1)/(4) Simplify the x terms using the rule (a^m)/(a^n) = a^m-n: (x^4)/(x^2) = x^4-2 = x^2 Simplify the y terms: y^-2y^-1 = y^-2 - (-1) = y^-2+1 = y^-1 Simplify the z terms: (z^1)/(z^-2) = z^1 - (-2) = z^1+2 = z^3 Combine these simplified terms: (1)/(4) x^2 y^-1 z^3 = (x^2 z^3)/(4y) Now, apply the exponent (1)/(2) to the entire simplified fraction. Remember that raising to the power of (1)/(2) is the same as taking the square root. ((x^2 z^3)/(4y))^(1)/(2) = ((x^2)^1)/(2) (z^3)^(1)/(2)(4)^(1)/(2) (y)^(1)/(2) Apply the power to each term: (x^2)^(1)/(2) = x^2 × (1)/(2) = x^1 = x (z^3)^(1)/(2) = z^(3)/(2) (4)^(1)/(2) = sqrt(4) = 2 (y)^(1)/(2) = sqrt(y) Combine these results: (x z^3)/(2)2 sqrt(y) (x z^3)/(2)2 sqrt(y) Step 2: Simplify -6x^0 + 4x^2 + 3x^2. Recall that any non-zero number raised to the power of 0 is 1, so x^0 = 1. Substitute x^0 = 1 into the expression: -6(1) + 4x^2 + 3x^2 -6 + 4x^2 + 3x^2 Combine the like terms (4x^2 and 3x^2): -6 + (4+3)x^2 -6 + 7x^2 It is standard practice to write terms with higher powers first: 7x^2 - 6 7x^2 - 6 Step 3: Simplify, and write as an Index notation: 3 × 3 × 3 × 3 × (1)/(3) × (1)/(3). Group the terms: (3 × 3 × 3 × 3) × ((1)/(3) × (1)/(3)) Write the first group in index notation: 3 × 3 × 3 × 3 = 3^4 Write the second group in index notation: (1)/(3) × (1)/(3) = ((1)/(3))^2 Recall that (1)/(a^n) = a^-n, so (1)/(3) = 3^-1. Therefore, ((1)/(3))^2 = (3^-1)^2 = 3^-1 × 2 = 3^-2. Now multiply the two parts: 3^4 × 3^-2 Use the rule a^m × a^n = a^m+n: 3^4 + (-2) = 3^4-2 = 3^2 3^2 Step 4: Evaluate ((13^0)/(5))^2. Recall that any non-zero number raised to the power of 0 is 1, so 13^0 = 1. Substitute 13^0 = 1 into the expression: ((1)/(5))^2 Apply the exponent to both the numerator and the denominator: (1^2)/(5^2) Calculate the squares: (1 × 1)/(5 × 5) = (1)/(25) (1)/(25) Send me the next one 📸