Math Problem Statement
8^(1/3 x+2)=2^(x+6) 36^(1/2 x+4)=6^(x+8) 2^x+2^(x+1)=3⋅2^x
Solution
Let's solve these equations one by one.
Equation 1:
Step 1: Express both sides in terms of base 2.
- Recall that , so we can write:
Step 2: Equating the exponents (since the bases are the same):
Simplifying the left-hand side:
This results in a true statement, meaning the equation holds for all values of . So, the solution is all real numbers.
Equation 2:
Step 1: Express both sides in terms of base 6.
- Recall that , so we can write:
Step 2: Equating the exponents (since the bases are the same):
Simplifying the left-hand side:
This is another true statement, meaning this equation also holds for all values of . So, the solution is all real numbers.
Equation 3:
Step 1: Factor out from the left-hand side:
This simplifies to:
Step 2: Divide both sides by (assuming ):
This is trivially true, meaning the equation holds for all values of . So, the solution is all real numbers.
Final Answer:
For all three equations:
- The solutions are all real numbers.
Would you like more detailed steps on any of these equations, or is there a specific concept you want to explore further?
Follow-up Questions:
- How do you approach solving exponential equations with different bases?
- How can we solve exponential equations involving fractional exponents?
- What is the general method for solving equations where exponents are linear expressions?
- What strategies can be used to solve equations with different logarithmic bases?
- How do we solve exponential equations when the bases cannot be easily rewritten in terms of each other?
Tip:
When dealing with exponential equations, always look for ways to rewrite both sides using the same base, as this simplifies the process of equating the exponents.
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Math Problem Analysis
Mathematical Concepts
Exponential Equations
Algebra
Exponents
Logarithms
Formulas
a^(m+n) = a^m * a^n
a^(m) = b^(n) implies m = n when a = b
Factoring expressions
Equating exponents for like bases
Theorems
Properties of exponents
Logarithmic and exponential equivalence
Suitable Grade Level
Grades 8-10
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