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.

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

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.

$$= -1$$ $$\boxed{\text{-1}}$$

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.

$$= -6$$ $$\boxed{\text{-6}}$$

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.

$$= 8$$ $$\boxed{\text{8}}$$

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:

$$$$