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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

Ask by Mathis Carter. in South Africa
Mar 09,2025

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**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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Exponent rules are foundational in mathematics and can boost your problem-solving skills! Firstly, the **Product Rule** states that when multiplying two powers with the same base, you add the exponents: \( a^m \cdot a^n = a^{m+n} \). Secondly, the **Quotient Rule** tells you that when dividing powers with the same base, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \). Finally, the **Power Rule** allows you to raise a power to another power by multiplying the exponents: \( (a^m)^n = a^{mn} \). Understanding these will help you navigate through exponent equations effortlessly! Now, let’s make these equations a bit spicier! For \( 5^{3x} - 5^{3x} = 4 \), notice the redundancy—simplify it down to \( 0 = 4 \), which screams contradiction. For \( 3^{x+1} \cdot 5 - 4 \cdot 3^{x+2} = -\frac{7}{3} \), you’ll want to isolate \( 3^x \) first, leading to a neat solution after factoring out the common terms. Get ready to play detective because solving these equations is like unlocking a treasure!

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