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Question 7.1 - Simplify Using the Laws of Exponents

7.1.1
$\frac{a^7b^4}{ab^3}$

Solution:
Apply the laws of exponents: $\frac{a^m}{a^n} = a^{m-n}$ and $\frac{b^m}{b^n} = b^{m-n}$ $= \frac{a^7}{a^1} \cdot \frac{b^4}{b^3} = a^{7-1} \cdot b^{4-3} = a^6b^1 = a^6b$

7.1.2
$\left( \frac{m^4n^2}{p^5} \right)^2$

Solution:
Apply the power of a quotient rule: $\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}$ $= \frac{(m^4)^2(n^2)^2}{(p^5)^2} = \frac{m^{4 \times 2} \cdot n^{2 \times 2}}{p^{5 \times 2}} = \frac{m^8 \cdot n^4}{p^{10}}$

Question 7.2 - Simplify Each of the Following

7.2.1
$a(1 - x) - a(1 + x)$

Solution:
Distribute $a$ across the terms inside the parentheses: $= a(1 - x) - a(1 + x) = a - ax - a - ax = -2ax$

7.2.2
$(3x^2 + x - 6) + (2x^2 - x + 5)$

Solution:
Combine like terms: $= (3x^2 + 2x^2) + (x - x) + (-6 + 5) = 5x^2 + 0x - 1 = 5x^2 - 1$

Question 7.3 - Factorize Each of the Following

7.3.1
$x^2 - x - 20$

Solution:
Factor by finding two numbers that multiply to $-20$ and add to $-1$: $x^2 - x - 20 = (x - 5)(x + 4)$

7.3.2
$2a^2 + 3a - 5$

Solution:
Factor by grouping or using the quadratic formula method. We look for factors of $2a^2$ and $-5$ that give us a middle term of $3a$.

7.4 - Solve for $x$ and $y$

2x - y &= 5 \\ x + 2y &= 3 \end{aligned}$$ **Solution**: Solve the system of linear equations using substitution or elimination. **7.5 - Solve the Following Inequality** $$3(x - 1) < 2(x + 4) - 7$$ **Solution**: Distribute and solve for $$x$$: $$3x - 3 < 2x + 8 - 7 \Rightarrow 3x - 3 < 2x + 1 \Rightarrow x < 4$$ **7.6 - Solve for the Roots** $$x^2 + 2x - 3 = 0$$ **Solution**: Factor or use the quadratic formula to find roots. **7.7 - Given that $$k = 1$$** For the line passing through $$A(2,5)$$ and $$B$$, solve the questions regarding slope, distance, midpoint, and parallelism. ### Would you like a more detailed explanation for any specific question? Here are five related questions: 1. How do you apply the laws of exponents to more complex expressions? 2. Can you explain the method for solving systems of linear equations? 3. What is the quadratic formula, and when should it be used? 4. How do you determine the slope of a line between two points? 5. What are some common mistakes to avoid when factorizing quadratic expressions? **Tip:** Always double-check your work, especially in multi-step problems, to avoid simple calculation errors.