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

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. $= 106 - 135$ $= -29$ $\boxed{-29}$

Question 4.2: Without using a calculator, calculate the following using the correct order of operations:

Question 4.2.1: $(8)(-4) \div (34-2)$ Step 1: Perform the operation inside the parentheses first. $= (8)(-4) \div (32)$ Step 2: Perform the multiplication. $= -32 \div 32$ Step 3: Perform the division. $= -1$ $\boxed{-1}$

Question 4.2.2: $\frac{\sqrt{64} + \sqrt{16}}{\sqrt[3]{27} - \sqrt[3]{125}}$ Step 1: Calculate the square roots in the numerator. $\sqrt{64} = 8$ $\sqrt{16} = 4$ Step 2: Calculate the cube roots in the denominator. $\sqrt[3]{27} = 3$ $\sqrt[3]{125} = 5$ Step 3: Substitute these values back into the expression. $= \frac{8 + 4}{3 - 5}$ Step 4: Perform the addition in the numerator and the subtraction in the denominator. $= \frac{12}{-2}$ Step 5: Perform the division. $= -6$ $\boxed{-6}$

Question 5: Simplify

Question 5.1: $9y^0 + (9y)^0 \times (-1)^9$ Step 1: Apply the exponent rule $a^0 = 1$ (for $a \neq 0$). Assume $y \neq 0$. $y^0 = 1$ $(9y)^0 = 1$ Step 2: Calculate $(-1)^9$. An odd power of $-1$ is $-1$. $(-1)^9 = -1$ Step 3: Substitute these values into the expression. $= 9(1) + 1 \times (-1)$ Step 4: Perform the multiplication. $= 9 - 1$ Step 5: Perform the subtraction. $= 8$ $\boxed{8}$

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 $(x^m)^n = x^{mn}$. $= -8 a^3 b^{2 \times 3}$ $= -8 a^3 b^6$ $\boxed{-8a^3b^6}$

Question 5.3: $\frac{-30x^{10}y^3}{3x^5y^2}$ Step 1: Divide the coefficients. $= \left(\frac{-30}{3}\right) x^{10}y^3 x^{-5}y^{-2}$ $= -10 x^{10}y^3 x^{-5}y^{-2}$ Step 2: Apply the exponent rule $x^m x^n = x^{m+n}$ for variables with the same base. $= -10 x^{10-5} y^{3-2}$ $= -10 x^5 y^1$ $= -10 x^5 y$ $\boxed{-10x^5y}$

Question 5.4: $\frac{3ab^6 \times (-3ab)^4}{3^3 a^5 b^{10}}$ 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 $3^3$. $= \frac{3ab^6 \times 81a^4b^4}{27 a^5 b^{10}}$ Step 3: Multiply the terms in the numerator. $= \frac{(3 \times 81) (a^1 \times a^4) (b^6 \times b^4)}{27 a^5 b^{10}}$ $= \frac{243 a^{1+4} b^{6+4}}{27 a^5 b^{10}}$ $= \frac{243 a^5 b^{10}}{27 a^5 b^{10}}$ Step 4: Divide the coefficients and apply the exponent rule $\frac{x^m}{x^n} = x^{m-n}$. $= \left(\frac{243}{27}\right) a^{5-5} b^{10-10}$ $= 9 a^0 b^0$ Step 5: Apply $a^0 = 1$. $= 9 \times 1 \times 1$ $= 9$ $\boxed{9}$

Question 5.5: $\frac{(-2x^2y^4)^2 \times 5xy^2}{\sqrt{100x^4y^{10}}}$ Step 1: Simplify the numerator. $(-2x^2y^4)^2 = (-2)^2 (x^2)^2 (y^4)^2 = 4x^4y^8$ Now multiply by $5xy^2$: $4x^4y^8 \times 5xy^2 = (4 \times 5) (x^4 \times x^1) (y^8 \times y^2)$ $= 20 x^{4+1} y^{8+2} = 20x^5y^{10}$ Step 2: Simplify the denominator. $\sqrt{100x^4y^{10}} = \sqrt{100} \times \sqrt{x^4} \times \sqrt{y^{10}}$ $= 10 \times x^{4/2} \times y^{10/2}$ $= 10x^2y^5$ Step 3: Combine the simplified numerator and denominator. $= \frac{20x^5y^{10}}{10x^2y^5}$ Step 4: Divide the coefficients and apply the exponent rule $\frac{x^m}{x^n} = x^{m-n}$. $= \left(\frac{20}{10}\right) x^{5-2} y^{10-5}$ $= 2x^3y^5$ $\boxed{2x^3y^5}$

Question 5.6: $\left(\frac{y^2x^{-1}}{y^2x y^2}\right)^{-2}$ 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: $\frac{y^2x^{-1}}{xy^4}$ Step 2: Apply the exponent rules for division ($\frac{x^m}{x^n} = x^{m-n}$). For $x$: $x^{-1-1} = x^{-2}$ For $y$: $y^{2-4} = y^{-2}$ So, the expression inside the parentheses simplifies to: