Separate the numerical coefficients and the variable terms.

Here are the solutions to the problems on Laws of Indices: 1. Simplify the expression (30x^4)/(3x^-1) Step 1: Separate the numerical coefficients and the variable terms. (30x^4)/(3x^-1) = ((30)/(3)) × ((x^4)/(x^-1)) Step 2: Simplify the numerical part. (30)/(3) = 10 Step 3: Simplify the variable part using the rule (a^m)/(a^n) = a^m-n. (x^4)/(x^-1) = x^4 - (-1) = x^4+1 = x^5 Step 4: Combine the simplified parts. 10 × x^5 = 10x^5 Answer: The simplified expression is 10x^5. 2. Simplify ((x^4)/(2y^2))^-2 Step 1: Apply the negative exponent rule ((a)/(b))^-n = ((b)/(a))^n. ((x^4)/(2y^2))^-2 = ((2y^2)/(x^4))^2 Step 2: Apply the power of a quotient rule ((a)/(b))^n = (a^n)/(b^n). ((2y^2)/(x^4))^2 = ((2y^2)^2)/((x^4)^2) Step 3: Apply the power of a product rule (ab)^n = a^n b^n and the power of a power rule (a^m)^n = a^mn. ((2y^2)^2)/((x^4)^2) = (2^2 (y^2)^2)/(x^4 × 2) = 4y^2 × 2x^8 = (4y^4)/(x^8) Answer: The simplified expression is (4y^4)/(x^8). 3. Simplify the powers: x^6 y^2 z^-3x^-1 y^2 z^-5 Step 1: Apply the division rule for exponents (a^m)/(a^n) = a^m-n to each variable separately. For x: (x^6)/(x^-1) = x^6 - (-1) = x^6+1 = x^7 For y: (y^2)/(y^2) = y^2-2 = y^0 = 1 For z: z^-3z^-5 = z^-3 - (-5) = z^-3+5 = z^2 Step 2: Multiply the simplified terms together. x^7 × 1 × z^2 = x^7 z^2 Answer: The simplified expression is x^7 z^2. Last free one today — make it count tomorrow, or type /upgrade for unlimited.