Matsavana Circuit Grade 11 Marhemarics Investlyation INSTRUCTIONS AND INFORMATION: Read the following instructions carefully before answering questions. 1. Complete ALL the investigations (\#1, \#2 and \#3) 2. Use spaces provided on each investigation sheet to answer the questions. 3. Write legibly and neatly for presentable work. INVESTIGATION \# 1 : 1.1 State the exponent rules and clearly explain how each rule works. $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ 1.2 In the following equations, solve for $$ x $$ (to two decimal places where necessary) and write all the steps until the final answer. a. $$ 5^{3 x}-5^{3 x} 1=4 $$ b. $$ 3^{x+1} \cdot 5-4 \cdot 3^{x+2}=-\frac{7}{3} $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ (4) $$ \qquad $$ (4) c. $$ 3^{2-x}-3^{-x-3}=\frac{242}{9} $$ d. $$ 52 x_{4} 4-25^{x} 1=78120 $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ (4) $$ \qquad $$ (5). Page 2 of 4

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Let's break down the investigation step by step.

### INVESTIGATION #1

#### 1.1 Exponent Rules

1. **Product of Powers Rule**: 
   $$
   a^m \cdot a^n = a^{m+n}
   $$
   - **Explanation**: When multiplying two powers with the same base, you add the exponents.

2. **Quotient of Powers Rule**: 
   $$
   \frac{a^m}{a^n} = a^{m-n}
   $$
   - **Explanation**: When dividing two powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator.

3. **Power of a Power Rule**: 
   $$
   (a^m)^n = a^{m \cdot n}
   $$
   - **Explanation**: When raising a power to another power, you multiply the exponents.

4. **Power of a Product Rule**: 
   $$
   (ab)^n = a^n \cdot b^n
   $$
   - **Explanation**: When raising a product to a power, you raise each factor to the power.

5. **Power of a Quotient Rule**: 
   $$
   \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
   $$
   - **Explanation**: When raising a quotient to a power, you raise both the numerator and the denominator to the power.

6. **Zero Exponent Rule**: 
   $$
   a^0 = 1 \quad (a \neq 0)
   $$
   - **Explanation**: Any non-zero number raised to the power of zero is equal to one.

7. **Negative Exponent Rule**: 
   $$
   a^{-n} = \frac{1}{a^n} \quad (a \neq 0)
   $$
   - **Explanation**: A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent.

---

#### 1.2 Solving for $$ x $$

**a.** $$ 5^{3x} - 5^{3x} = 4 $$

This equation seems to have a typo. It should likely be $$ 5^{3x} - 5^{3x-1} = 4 $$. Let's solve it assuming that.

1. Rewrite the equation:
   $$
   5^{3x} - \frac{5^{3x}}{5} = 4
   $$
   $$
   5^{3x} \left(1 - \frac{1}{5}\right) = 4
   $$
   $$
   5^{3x} \cdot \frac{4}{5} = 4
   $$
2. Multiply both sides by $$ \frac{5}{4} $$:
   $$
   5^{3x} = 5
   $$
3. Rewrite $$ 5 $$ as $$ 5^1 $$:
   $$
   5^{3x} = 5^1
   $$
4. Set the exponents equal to each other:
   $$
   3x = 1
   $$
5. Solve for $$ x $$:
   $$
   x = \frac{1}{3} \approx 0.33
   $$

---

**b.** $$ 3^{x+1} \cdot 5 - 4 \cdot 3^{x+2} = -\frac{7}{3} $$

1. Rewrite the equation:
   $$
   3^{x+1} \cdot 5 - 4 \cdot 3^{x+1} \cdot 3 = -\frac{7}{3}
   $$
   $$
   3^{x+1} \cdot (5 - 12) = -\frac{7}{3}
   $$
   $$
   3^{x+1} \cdot (-7) = -\frac{7}{3}
   $$
2. Divide both sides by -7:
   $$
   3^{x+1} = \frac{1}{3}
   $$
3. Rewrite $$ \frac{1}{3} $$ as $$ 3^{-1} $$:
   $$
   3^{x+1} = 3^{-1}
   $$
4. Set the exponents equal to each other:
   $$
   x + 1 = -1
   $$
5. Solve for $$ x $$:
   $$
   x = -2
   $$

---

**c.** $$ 3^{2-x} - 3^{-x-3} = \frac{242}{9} $$

1. Rewrite $$ 3^{-x-3} $$ as $$ \frac{1}{3^{x+3}} $$:
   $$
   3^{2-x} - \frac{1}{3^{x+3}} = \frac{242}{9}
   $$
2. Multiply through by $$ 3^{x+3} $$ to eliminate the fraction:
   $$
   3^{(2-x)+(x+3)} - 1 = \frac{242 \cdot 3^{x+3}}{9}
   $$
   $$
   3^5 - 1 = \frac{242 \cdot 3^{x+3}}{9}
   $$
3. Calculate $$ 3^5 = 243 $$:
   $$
   242 = \frac{242 \cdot 3^{x+3}}{9}
   $$
4. Multiply both sides by 9:
   $$
   2178 = 242 \cdot 3^{x+3}
   $$
5. Divide both sides by 242:
   $$
   3^{x+3} = \frac{2178}{242} \approx 9
   $$
6. Rewrite $$ 9 $$ as $$ 3^2 $$:
   $$
   3^{x+3} = 3^2
   $$
7. Set the exponents equal to each other:
   $$
   x + 3 = 2
   $$
8. Solve for $$ x $$:
   $$
   x = -1
   $$

---

**d.** $$ 5^{2x} - 25^{x} = 78120 $$

1. Rewrite $$ 25^x $$ as $$ (5^2)^x = 5^{2x} $$:
   $$
   5^{2x} - 5^{2x} = 78120
   $$
   This equation is incorrect as it simplifies to $$ 0 = 78120 $$. Please check the equation for any errors.

---

Please review the calculations and let me know if you need further assistance or corrections!
**INVESTIGATION #1** 1. **Exponent Rules:** - **Product of Powers:** $$ a^m \cdot a^n = a^{m+n} $$ - **Quotient of Powers:** $$ \frac{a^m}{a^n} = a^{m-n} $$ - **Power of a Power:** $$ (a^m)^n = a^{m \cdot n} $$ - **Power of a Product:** $$ (ab)^n = a^n \cdot b^n $$ - **Power of a Quotient:** $$ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $$ - **Zero Exponent:** $$ a^0 = 1 $$ (where $$ a \neq 0 $$) - **Negative Exponent:** $$ a^{-n} = \frac{1}{a^n} $$ (where $$ a \neq 0 $$) 2. **Solutions for $$ x $$:** - **a.** $$ x \approx 0.33 $$ - **b.** $$ x = -2 $$ - **c.** $$ x = -1 $$ - **d.** The equation is inconsistent and has no solution. Please review the calculations and let me know if you need further assistance!

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