#matholympiad #mathequations #exponentialequations
✅ Video Overview:
In this session, we tackle a complex exponential math olympiad problem equation and break it down into manageable steps. We'll explore the properties of exponents, apply algebraic manipulations, and arrive at the correct values of x. This video is perfect for students preparing for competitive exams, math enthusiasts, and anyone looking to enhance their problem-solving skills.
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✅ Key Concepts Covered:
1. Properties of Exponents:
2. Algebraic Manipulations:
3. Step-by-Step Problem Solving:
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✅ Detailed Steps:
Problem:
(8^x + 27^x) / (12^x + 18^x) = 7 / 6
Step 1: Express 8, 27, 12, and 18 in terms of their prime factors:
8 = 2^3, 27 = 3^3, 12 = 2^2 * 3, 18 = 2 * 3^2
Step 2: Substitute these values into the equation:
(2^3)^x + (3^3)^x / (2^2 * 3)^x + (2 * 3^2)^x = 7 / 6
Step 3: Simplify the equation using the properties of exponents:
2^(3x) + 3^(3x) / (2^2)^x * 3^x + 2^x * (3^2)^x = 7 / 6
2^(3x) + 3^(3x) / 2^(2x) * 3^x + 2^x * 3^(2x) = 7 / 6
Step 4: Let a = 2^x and b = 3^x, then the equation becomes:
(a^3 + b^3) / (a^2 * b + a * b^2) = 7 / 6
Step 5: Factorize and simplify:
(a + b)(a^2 - ab + b^2) / ab(a + b) = 7 / 6
(a^2 - ab + b^2) / ab = 7 / 6
Step 6: Multiply both sides by 6ab:
6(a^2 - ab + b^2) = 7ab
Step 7: Simplify the equation:
6a^2 - 13ab + 6b^2 = 0
Step 8: Factorize the quadratic equation:
(2a - 3b)(3a - 2b) = 0
Step 9: Solve the resulting equations:
2a = 3b or 3a = 2b
2(2^x) = 3(3^x) or 3(2^x) = 2(3^x)
Step 10: Solve for x:
2^(x+1) = 3^(x+1) or 3 * 2^x = 2 * 3^x
x = -1 or x = 1
Thus, the solutions are:
x = -1, 1
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✅ Learning Outcomes:
By the end of this video, you will:
- Understand how to manipulate and simplify expressions involving exponents.
- Be able to apply algebraic properties to solve complex exponential equations.
- Gain confidence in breaking down and solving algebraic problems step-by-step.
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✅ Also you can see:
• The Secret to Solving ...
• Master the Pythagorean...
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✅ Tags:
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