Exponential Equations - Algebra and Precalculus
TLDRThis video script guides viewers through solving an exponential equation: \(3^{x+2} + 3^x = \frac{10}{81}\). It begins by introducing the properties of exponents, such as the rule for multiplying with the same base where exponents are added. The equation is then simplified by expressing \(3^{x+2}\) as \(3^2 \cdot 3^x\) and rewriting the denominator 81 as \(3^4\). The script demonstrates combining like terms and manipulating the equation to isolate \(3^x\), resulting in \(3^x = 3^{-4}\). The solution is found by equating the exponents since the bases are the same, yielding \(x = -4\). The script concludes by verifying the solution by substituting \(x\) back into the original equation, confirming the correctness of \(x = -4\).
Takeaways
- 🧩 The problem presented is an exponential equation: 3^{x+2} + 3^x = \frac{10}{81}.
- 🔍 The script emphasizes the importance of understanding exponent properties, such as multiplying exponents with the same base.
- 📚 It explains that x^4 \cdot x^5 = x^{4+5} = x^9, and the reverse process can be applied to break down exponents.
- 📝 The term 3^{x+2} is rewritten as 3^x \cdot 3^2 to simplify the equation.
- 🔢 The number 81 is expressed as 3^4 to align the bases on both sides of the equation.
- ➕ The equation simplifies to 9 \cdot 3^x + 3^x = \frac{10}{3^{-4}} after recognizing 3^2 = 9.
- 🤔 The script prompts for the next logical step, which involves combining like terms on the left side of the equation.
- 📉 The coefficients of 3^x are added together (9 + 1 = 10) to simplify further.
- 🔄 The script explains the process of moving the base from the denominator to the numerator, changing the exponent's sign.
- 📈 After dividing both sides by 10, the equation becomes 3^x = 3^{-4}, indicating that the bases are equal and thus the exponents must be equal.
- 🎯 The conclusion is that x = -4, which is verified by substituting back into the original equation to ensure correctness.
- 📚 The script concludes with a demonstration of how to verify the solution by plugging x = -4 back into the original equation.
Q & A
What is the original exponential equation presented in the video?
-The original exponential equation is \(3^{x+2} + 3^x = \frac{10}{81}\).
What property of exponents is mentioned at the beginning of the video?
-The property of exponents mentioned is that when you multiply with a common base, you can add the exponents.
How does the video suggest rewriting the term \(3^{x+2}\)?
-The video suggests rewriting \(3^{x+2}\) as \(3^x \times 3^2\) because it's equivalent to \(3^{x+2}\).
Why does the video recommend changing 81 into a base of 3?
-The video recommends changing 81 into a base of 3 because 81 is a power of 3 (\(3^4 = 81\)) which simplifies the equation by having a common base.
What algebraic operation is performed on the left side of the equation after rewriting it?
-The algebraic operation performed is factoring out the common base, which is \(3^x\), resulting in \(9 \times 3^x + 3^x\).
How does the video simplify the expression \(9 \times 3^x + 3^x\)?
-The video simplifies it by combining like terms, adding the coefficients 9 and 1 to get \(10 \times 3^x\).
What rule is applied when moving the 3 from the denominator to the numerator?
-The rule applied is that when you move a base from the bottom to the top of a fraction, the exponent changes sign, resulting in \(3^{-4}\).
What does the video suggest doing after moving the 3 to the numerator?
-The video suggests dividing both sides of the equation by 10 to isolate \(3^x\) on one side.
How does the video conclude that \(x\) is equal to negative four?
-The video concludes that \(x\) is equal to negative four by setting \(3^x\) equal to \(3^{-4}\) and recognizing that if the bases are the same, the exponents must be equal.
What verification method does the video use to ensure the correctness of the solution?
-The video verifies the solution by plugging \(x = -4\) back into the original equation and checking if both sides are equal.
Outlines
🔍 Solving an Exponential Equation
The video script begins by presenting an exponential equation: \(3^{x+2} + 3^x = \frac{10}{81}\). The speaker encourages viewers to attempt solving the problem before revealing the solution. The process involves understanding the properties of exponents, such as how to multiply and divide them. The equation is rewritten by expressing \(3^{x+2}\) as \(3^2 \cdot 3^x\), simplifying the left side to \(9 \cdot 3^x + 3^x\). Recognizing that 81 is \(3^4\), the equation is transformed to \(10 \cdot 3^x = 3^{-4}\). The next step is to combine terms on the left side, resulting in \(10 \cdot 3^x = 3^4 \cdot 3^{-4}\). The video explains the rule of changing the sign of the exponent when moving from the numerator to the denominator, leading to \(3^x = 3^{-4}\). Since the bases are the same, the exponents must be equal, concluding that \(x = -4\). The speaker then verifies the solution by substituting \(x\) with \(-4\) in the original equation.
📚 Verification of the Exponential Equation Solution
The second paragraph of the script continues with the verification process of the solution \(x = -4\). The speaker substitutes \(x\) with \(-4\) in the original equation and simplifies the expression step by step. The term \(3^{x+2}\) becomes \(3^{-2} + 2\), which simplifies to \(\frac{1}{9} + 2\). The term \(3^x\) is \(3^{-4}\), which is \(\frac{1}{81}\). The script then shows the calculation of \(3^2 \cdot 3^{-4}\), which simplifies to \(\frac{9}{81}\) or \(\frac{1}{9}\). The fractions are combined by adding the numerators over the common denominator, resulting in \(\frac{10}{81}\), which matches the right side of the original equation. This confirms that the solution \(x = -4\) is correct. The video concludes with a summary of how the problem was solved and thanks the viewers for watching.
Mindmap
Keywords
💡Exponential Equation
💡Exponent
💡Base
💡Coefficient
💡Negative Exponent
💡Property of Exponents
💡Simplification
💡Substitution
💡Verification
💡Fraction
Highlights
Introduction to the problem: solving the exponential equation 3^(x+2) + 3^x = 10/81.
Explanation of the property of exponents: when multiplying with a common base, add the exponents.
Rewriting 3^(x+2) as 3^x * 3^2 to simplify the equation.
Replacing 81 with 3^4 since 81 is a multiple of three.
Simplifying 9 * 3^x + 3^x by factoring out 3^x.
Combining like terms: 9 * 3^x + 3^x becomes 10 * 3^x.
Using the property of negative exponents: moving 3^4 from the denominator to the numerator changes the sign of the exponent.
Equating the exponents since the bases are the same: 3^x = 3^-4 implies x = -4.
Verification step: substituting x = -4 back into the original equation to check the solution.
Calculation of 3^(-4+2) and 3^-4, verifying that the sum equals 10/81.
Explanation of negative exponents: x^-3 is the same as 1/x^3.
Combining fractions with common denominators to simplify verification.
Final confirmation that the left side equals the right side, validating x = -4 as the correct solution.
Conclusion: summarizing the steps to solve the exponential equation.
Encouragement to practice similar problems and apply the properties of exponents.
Transcripts
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