Problem 19 - College Algebra Essentials

Question

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$8^{x+3}=16^{x-1}$$

Step-by-Step Solution

Answer

The solution to the equation \(8^{x+3} = 16^{x-1}\) is \(x = 13\).

Step 1 - Express each side as a power of the same base

Find a common base for both 8 and 16. Since both of these can be expressed as powers of 2, we can write: \(8=2^3\) and \(16=2^4\). So the given expression can be rewritten as: \((2^3)^{x+3}=(2^4)^{x-1}\)

Step 2 - Utilize the power of a power rule

From the power of a power rule, we know that \((a^m)^n = a^{mn}\). Using this rule for both sides, we can simplify to: \(2^{3(x+3)} = 2^{4(x-1)}\)

Step 3 - Equate exponents

Since the base on both sides of the equation is the same, we can equate the exponents and solve the resulting equation: \(3(x+3) = 4(x-1)\)

Step 4 - Simplify and solve

Simplify to get an equation in terms of \(x\): \(3x+9 = 4x-4\). By transposing, we get \(x = 13\)