Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 8^(x+3)=16^(x−1)

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Identify the bases of the exponential expressions: 8 and 16.

Express each base as a power of a common base. Note that 8 = 2^3 and 16 = 2^4.

Rewrite the equation using these expressions: (2^3)^(x+3) = (2^4)^(x−1).

Apply the power of a power property: 2^(3(x+3)) = 2^(4(x−1)).

Since the bases are the same, set the exponents equal to each other: 3(x+3) = 4(x−1).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential Equations

Exponential equations are mathematical expressions where variables appear in the exponent. To solve these equations, one common method is to express both sides with the same base, allowing for the exponents to be equated. This approach simplifies the equation and makes it easier to isolate the variable.

Base Conversion

Base conversion involves rewriting numbers or expressions in terms of a common base. In the context of the given equation, both 8 and 16 can be expressed as powers of 2 (8 = 2^3 and 16 = 2^4). This conversion is crucial for equating the exponents and solving the equation effectively.

Equating Exponents

Equating exponents is a technique used after both sides of an exponential equation have been expressed with the same base. Once the bases are the same, the exponents can be set equal to each other, leading to a simpler algebraic equation. This step is essential for finding the value of the variable in the original equation.

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