Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 6^(x−3)/4=√6

Verified step by step guidance

1. Recognize that the equation is \(6^{\frac{x-3}{4}} = \sqrt{6}\). The goal is to express both sides as powers of the same base.
2. Rewrite the right side, \(\sqrt{6}\), as a power of 6. Recall that \(\sqrt{6} = 6^{\frac{1}{2}}\).
3. Now the equation becomes \(6^{\frac{x-3}{4}} = 6^{\frac{1}{2}}\).
4. Since the bases are the same and the equation is an equality, set the exponents equal to each other: \(\frac{x-3}{4} = \frac{1}{2}\).
5. Solve the resulting linear equation for \(x\) by multiplying both sides by 4 and then isolating \(x\).

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent position. Solving these requires understanding how to manipulate and isolate the variable in the exponent to find its value.

Expressing Numbers as Powers of the Same Base

To solve exponential equations, rewrite both sides with the same base if possible. This allows you to set the exponents equal to each other, simplifying the equation to a solvable form.

Properties of Exponents

Properties such as the power of a power and the product of powers help simplify expressions. For example, the square root of a number can be written as that number raised to the 1/2 power, aiding in rewriting terms with common bases.