91Ó°ÊÓ

Understanding the properties of exponents is fundamental when dealing with exponential equations. Exponents are a shorthand way to express repeated multiplication of a base number.
For example, in the equation \((3.2)^{x-3} = (3.2)^{5}\), the base is 3.2, and the exponents are \(x-3\) and 5.
Here are some key properties:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Power of a Power: \( (a^m)^n = a^{m \times n} \)
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)
• Zero Exponent: \( a^0 = 1 \) for any \(a eq 0 \)

In the given problem, we used a property called \(a^m = a^n \rightarrow m = n\) when the bases are the same, allowing us to equate the exponents directly.

When solving any type of equation, the main goal is to find the unknown variable. In exponential equations, it’s often useful to isolate the exponents, especially if the base is the same on both sides, like in this problem.
Here’s the systematic approach we used:
1. **Identify the Base:** Recognize that both sides of the equation have the base 3.2.
2. **Set Exponents Equal:** Using the property mentioned earlier, set the exponents equal when the bases match: \(x - 3 = 5\)
3. **Simplify the Equation:** Perform algebraic operations to isolate the unknown variable.
This method reduces complex problems to simpler linear equations, making them easier to solve.

Isolation of variables is a core concept that allows us to solve for the unknown variable step by step.
Here’s how we apply this in our example:
1. **Set Up the Equation:** From solving the exponents, we get \(x - 3 = 5\)
2. **Isolate the Variable:** Move terms involving the variable to one side and constants to the other. For \(x - 3 = 5\), add 3 to both sides:
\[ x - 3 + 3 = 5 + 3 \]
\[ x = 8 \]
3. **Verify Your Solution:** Always substitute the solved value back into the original equation to check. In this case, substituting \ x = 8 \ confirms the equation \ (3.2)^{8-3} = (3.2)^{5} \ holds true.

This logical step-by-step isolation of variables ensures accuracy and makes even complex equations manageable.