Simplify the following expressions: 5.1 9y⁰ + (9y)⁰ × (-1)⁹, 5.2 (-2ab²)³, 5.3 (-30x¹⁰y³)/(3x⁵y²), 5.4 (3ab⁶x - (-3ab)⁴)/(3³a⁵b¹⁰), 5.5 ((-2x²y⁴)² × 5xy²)/√(100x⁴y¹⁰), 5.6 ((y² + 1/y⁻²)/(y²x/y⁻²))⁻²

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Asked on March 24, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-03-24T20:35:42.369Z

Question 4.1: By using the commutative and associative properties, calculate: -33 + 103 - 102 + 3 Step 1: Rearrange the terms using the commutative property (a+b=b+a) to group positive and negative numbers, or numbers that simplify easily. = (103 + 3) + (-33 - 102) Step 2: Perform the additions within the parentheses. = 106 + (-135) Step 3: Perform the final subtraction. = 106 - 135 = -29 -29 Question 4.2: Without using a calculator, calculate the following using the correct order of operations: Question 4.2.1: (8)(-4) ÷ (34-2) Step 1: Perform the operation inside the parentheses first. = (8)(-4) ÷ (32) Step 2: Perform the multiplication. = -32 ÷ 32 Step 3: Perform the division. = -1 -1 Question 4.2.2: sqrt(64) + sqrt(16)[3]27 - [3]125 Step 1: Calculate the square roots in the numerator. sqrt(64) = 8 sqrt(16) = 4 Step 2: Calculate the cube roots in the denominator. [3]27 = 3 [3]125 = 5 Step 3: Substitute these values back into the expression. = (8 + 4)/(3 - 5) Step 4: Perform the addition in the numerator and the subtraction in the denominator. = (12)/(-2) Step 5: Perform the division. = -6 -6 Question 5: Simplify Question 5.1: 9y^0 + (9y)^0 × (-1)^9 Step 1: Apply the exponent rule a^0 = 1 (for a ≠ 0). Assume y ≠ 0. y^0 = 1 (9y)^0 = 1 Step 2: Calculate (-1)^9. An odd power of -1 is -1. (-1)^9 = -1 Step 3: Substitute these values into the expression. = 9(1) + 1 × (-1) Step 4: Perform the multiplication. = 9 - 1 Step 5: Perform the subtraction. = 8 8 Question 5.2: (-2ab^2)^3 Step 1: Apply the exponent rule (xyz)^n = x^n y^n z^n. = (-2)^3 a^3 (b^2)^3 Step 2: Calculate (-2)^3 and apply the exponent rule (x^m)^n = x^mn. = -8 a^3 b^2 × 3 = -8 a^3 b^6 -8a^3b^6 Question 5.3: -30x^10y^33x^5y^2 Step 1: Divide the coefficients. = ((-30)/(3)) x^10y^3 x^-5y^-2 = -10 x^10y^3 x^-5y^-2 Step 2: Apply the exponent rule x^m x^n = x^m+n for variables with the same base. = -10 x^10-5 y^3-2 = -10 x^5 y^1 = -10 x^5 y -10x^5y Question 5.4: (3ab^6 × (-3ab)^4)/(3^3 a^5 b^10) Step 1: Simplify the term (-3ab)^4 using (xyz)^n = x^n y^n z^n. (-3ab)^4 = (-3)^4 a^4 b^4 = 81 a^4 b^4 Step 2: Substitute this back into the expression and simplify 3^3. = (3ab^6 × 81a^4b^4)/(27 a^5 b^10) Step 3: Multiply the terms in the numerator. = ((3 × 81) (a^1 × a^4) (b^6 × b^4))/(27 a^5 b^10) = 243 a^1+4 b^6+427 a^5 b^10 = 243 a^5 b^1027 a^5 b^10 Step 4: Divide the coefficients and apply the exponent rule (x^m)/(x^n) = x^m-n. = ((243)/(27)) a^5-5 b^10-10 = 9 a^0 b^0 Step 5: Apply a^0 = 1. = 9 × 1 × 1 = 9 9 Question 5.5: ((-2x^2y^4)^2 × 5xy^2)/(sqrt(100x^4y^10)) Step 1: Simplify the numerator. (-2x^2y^4)^2 = (-2)^2 (x^2)^2 (y^4)^2 = 4x^4y^8 Now multiply by 5xy^2: 4x^4y^8 × 5xy^2 = (4 × 5) (x^4 × x^1) (y^8 × y^2) = 20 x^4+1 y^8+2 = 20x^5y^10 Step 2: Simplify the denominator. sqrt(100x^4y^10) = sqrt(100) × sqrt(x^4) × sqrt(y^10) = 10 × x^4/2 × y^10/2 = 10x^2y^5 Step 3: Combine the simplified numerator and denominator. = 20x^5y^1010x^2y^5 Step 4: Divide the coefficients and apply the exponent rule (x^m)/(x^n) = x^m-n. = ((20)/(10)) x^5-2 y^10-5 = 2x^3y^5 2x^3y^5 Question 5.6: (y^2x^-1y^2x y^2)^-2 Step 1: Simplify the expression inside the parentheses. Combine the terms in the denominator: y^2x y^2 = x y^2+2 = x y^4. The fraction becomes: y^2x^-1xy^4 Step 2: Apply the exponent rules for division ((x^m)/(x^n) = x^m-n). For x: x^-1-1 = x^-2 For y: y^2-4 = y^-2 So, the expression inside the parentheses simplifies to:

Accepted Answer · 2026-03-24T20:35:42.369Z

Question 4.1: By using the commutative and associative properties, calculate: -33 + 103 - 102 + 3 Step 1: Rearrange the terms using the commutative property (a+b=b+a) to group positive and negative numbers, or numbers that simplify easily. = (103 + 3) + (-33 - 102) Step 2: Perform the additions within the parentheses. = 106 + (-135) Step 3: Perform the final subtraction. = 106 - 135 = -29 -29 Question 4.2: Without using a calculator, calculate the following using the correct order of operations: Question 4.2.1: (8)(-4) ÷ (34-2) Step 1: Perform the operation inside the parentheses first. = (8)(-4) ÷ (32) Step 2: Perform the multiplication. = -32 ÷ 32 Step 3: Perform the division. = -1 -1 Question 4.2.2: sqrt(64) + sqrt(16)[3]27 - [3]125 Step 1: Calculate the square roots in the numerator. sqrt(64) = 8 sqrt(16) = 4 Step 2: Calculate the cube roots in the denominator. [3]27 = 3 [3]125 = 5 Step 3: Substitute these values back into the expression. = (8 + 4)/(3 - 5) Step 4: Perform the addition in the numerator and the subtraction in the denominator. = (12)/(-2) Step 5: Perform the division. = -6 -6 Question 5: Simplify Question 5.1: 9y^0 + (9y)^0 × (-1)^9 Step 1: Apply the exponent rule a^0 = 1 (for a ≠ 0). Assume y ≠ 0. y^0 = 1 (9y)^0 = 1 Step 2: Calculate (-1)^9. An odd power of -1 is -1. (-1)^9 = -1 Step 3: Substitute these values into the expression. = 9(1) + 1 × (-1) Step 4: Perform the multiplication. = 9 - 1 Step 5: Perform the subtraction. = 8 8 Question 5.2: (-2ab^2)^3 Step 1: Apply the exponent rule (xyz)^n = x^n y^n z^n. = (-2)^3 a^3 (b^2)^3 Step 2: Calculate (-2)^3 and apply the exponent rule (x^m)^n = x^mn. = -8 a^3 b^2 × 3 = -8 a^3 b^6 -8a^3b^6 Question 5.3: -30x^10y^33x^5y^2 Step 1: Divide the coefficients. = ((-30)/(3)) x^10y^3 x^-5y^-2 = -10 x^10y^3 x^-5y^-2 Step 2: Apply the exponent rule x^m x^n = x^m+n for variables with the same base. = -10 x^10-5 y^3-2 = -10 x^5 y^1 = -10 x^5 y -10x^5y Question 5.4: (3ab^6 × (-3ab)^4)/(3^3 a^5 b^10) Step 1: Simplify the term (-3ab)^4 using (xyz)^n = x^n y^n z^n. (-3ab)^4 = (-3)^4 a^4 b^4 = 81 a^4 b^4 Step 2: Substitute this back into the expression and simplify 3^3. = (3ab^6 × 81a^4b^4)/(27 a^5 b^10) Step 3: Multiply the terms in the numerator. = ((3 × 81) (a^1 × a^4) (b^6 × b^4))/(27 a^5 b^10) = 243 a^1+4 b^6+427 a^5 b^10 = 243 a^5 b^1027 a^5 b^10 Step 4: Divide the coefficients and apply the exponent rule (x^m)/(x^n) = x^m-n. = ((243)/(27)) a^5-5 b^10-10 = 9 a^0 b^0 Step 5: Apply a^0 = 1. = 9 × 1 × 1 = 9 9 Question 5.5: ((-2x^2y^4)^2 × 5xy^2)/(sqrt(100x^4y^10)) Step 1: Simplify the numerator. (-2x^2y^4)^2 = (-2)^2 (x^2)^2 (y^4)^2 = 4x^4y^8 Now multiply by 5xy^2: 4x^4y^8 × 5xy^2 = (4 × 5) (x^4 × x^1) (y^8 × y^2) = 20 x^4+1 y^8+2 = 20x^5y^10 Step 2: Simplify the denominator. sqrt(100x^4y^10) = sqrt(100) × sqrt(x^4) × sqrt(y^10) = 10 × x^4/2 × y^10/2 = 10x^2y^5 Step 3: Combine the simplified numerator and denominator. = 20x^5y^1010x^2y^5 Step 4: Divide the coefficients and apply the exponent rule (x^m)/(x^n) = x^m-n. = ((20)/(10)) x^5-2 y^10-5 = 2x^3y^5 2x^3y^5 Question 5.6: (y^2x^-1y^2x y^2)^-2 Step 1: Simplify the expression inside the parentheses. Combine the terms in the denominator: y^2x y^2 = x y^2+2 = x y^4. The fraction becomes: y^2x^-1xy^4 Step 2: Apply the exponent rules for division ((x^m)/(x^n) = x^m-n). For x: x^-1-1 = x^-2 For y: y^2-4 = y^-2 So, the expression inside the parentheses simplifies to:

Simplify the following expressions: 5.1 9y⁰ + (9y)⁰ × (-1)⁹, 5.2 (-2ab²)³, 5.3 (-30x¹⁰y³)/(3x⁵y²), 5.4 (3ab⁶x - (-3ab)⁴)/(3³a⁵b¹⁰), 5.5 ((-2x²y⁴)² × 5xy²)/√(100x⁴y¹⁰), 5.6 ((y² + 1/y⁻²)/(y²x/y⁻²))⁻²

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Simplify the following expressions: 5.1 9y⁰ + (9y)⁰ × (-1)⁹, 5.2 (-2ab²)³, 5.3 (-30x¹⁰y³)/(3x⁵y²), 5.4 (3ab⁶x - (-3ab)⁴)/(3³a⁵b¹⁰), 5.5 ((-2x²y⁴)² × 5xy²)/√(100x⁴y¹⁰), 5.6 ((y² + 1/y⁻²)/(y²x/y⁻²))⁻²

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