Find All Complex Solutions 2^(x+3)+2^x=72
The problem presents an equation involving exponential terms, namely 2^(x+3) and 2^x, and requires solving for the variable x, which is presumably in the complex number domain. You are asked to find all complex solutions that satisfy the equation. This involves the use of algebraic manipulation, properties of exponents, and potentially complex number arithmetic to isolate and solve for the variable x, yielding solutions that can be real, imaginary, or a combination of both (complex).
$2^{x + 3} + 2^{x} = 72$
Answer
Solution:
Step 1:
Extract $2^x$ as a common factor from the given expression to get $2^x(2^3 + 1)$.
Step 2:
Combine the terms inside the parentheses: $2^x \cdot (8 + 1)$.
Step 3:
Recognize that $8 + 1$ equals $9$, so rewrite the equation as $2^x \cdot 9 = 72$.
Step 4:
Isolate $2^x$ by dividing both sides of the equation by $9$.
Step 4.1:
Perform the division: $\frac{2^x \cdot 9}{9} = \frac{72}{9}$.
Step 4.2:
Simplify the equation by removing the common factor of $9$.
Step 4.2.1:
Eliminate the $9$s on the left side: $\frac{\cancel{9} \cdot 2^x}{\cancel{9}} = \frac{72}{9}$.
Step 4.2.1.1:
The equation simplifies to $2^x = \frac{72}{9}$.
Step 4.3:
Simplify the right side of the equation by dividing $72$ by $9$ to get $2^x = 8$.
Step 5:
Express $8$ as a power of $2$ to match the base on the left side: $2^x = 2^3$.
Step 6:
Since the bases are equal, set the exponents equal to each other to find the solution: $x = 3$.
Knowledge Notes:
The problem involves solving for $x$ in the equation $2^{x+3} + 2^x = 72$, where $x$ is a complex number. The solution process uses several algebraic techniques:
Factoring: The first step is to factor out the common term $2^x$ from the two terms on the left side of the equation. This is a standard technique for simplifying expressions.
Simplification: After factoring, the expression inside the parentheses is simplified by adding the constants together.
Isolation: The next step is to isolate the term with the variable ($2^x$) on one side of the equation. This is done by dividing both sides of the equation by the constant that is multiplied with $2^x$.
Cancellation: When dividing both sides by a common factor, the factor cancels out on the side where it is multiplied with the variable term, simplifying the equation further.
Exponentiation: Recognizing that both sides of the equation can be expressed as powers of the same base allows us to set the exponents equal to each other. This is based on the property that if $a^m = a^n$, then $m = n$ when $a$ is not equal to zero.
Solving for the Variable: Once the exponents are set equal to each other, the variable can be solved directly.
This problem assumes knowledge of exponent rules, factoring, and basic algebraic manipulation. It also relies on the understanding that an equation with equal bases can be solved by setting the exponents equal to each other.
