Question

Simplify the following expressions without the use of a calculator (show ALL
your working out and leave ALL answers with POSITIVE exponents).
2.1.1 (2)
3
2𝑥−9
3
𝑥−3
2.1.2 (4)
12+4 75
( 5+ 3)( 5− 3)

Answer 1

Let's simplify the expressions provided step by step.

### Problem 2.1.1:
**Expression:** \( \frac{2x^{-9}}{3x^{-3}} \)

**Step 1:** Rewrite each term in the expression to eliminate negative exponents:
\[
x^{-9} = \frac{1}{x^9} \quad \text{and} \quad x^{-3} = \frac{1}{x^3}
\]
So we can rewrite the expression:
\[
\frac{2x^{-9}}{3x^{-3}} = \frac{2}{3} \cdot \frac{1}{x^9} \cdot x^3 = \frac{2}{3} \cdot \frac{x^3}{x^9}
\]

**Step 2:** Apply the laws of exponents:
\[
\frac{x^3}{x^9} = x^{3-9} = x^{-6}
\]
Thus, substituting back, we have:
\[
\frac{2}{3} \cdot x^{-6} = \frac{2}{3x^6}
\]
**Final answer for 2.1.1:**
\[
\frac{2}{3x^6}
\]

### Problem 2.1.2:
**Expression:** \( \frac{12 + 4 \cdot 75}{(5+3)(5-3)} \)

**Step 1:** Calculate the denominator:
\[
(5 + 3)(5 - 3) = 8 \cdot 2 = 16
\]

**Step 2:** Calculate the numerator:
First, simplify \( 4 \cdot 75 \):
\[
4 \cdot 75 = 300
\]
Now, add to 12:
\[
12 + 300 = 312
\]

**Step 3:** Now put everything together:
\[
\frac{12 + 4 \cdot 75}{(5+3)(5-3)} = \frac{312}{16}
\]

**Step 4:** Simplify \( \frac{312}{16} \):
Divide both the numerator and the denominator by 16:
\[
312 \div 16 = 19.5 \quad \text{(but we will keep it in a proper fraction form)}
\]

**Step 5:** Converting to a proper simplified fraction:
Both numbers can still be divided by 4:
\[
\frac{312 \div 4}{16 \div 4} = \frac{78}{4}
\]
Again, dividing by 2:
\[
\frac{78 \div 2}{4 \div 2} = \frac{39}{2}
\]

**Final answer for 2.1.2:**
\[
\frac{39}{2}
\]

### Summary of Answers:
- **2.1.1:** \( \frac{2}{3x^6} \)
- **2.1.2:** \( \frac{39}{2} \)