91Ó°ÊÓ

The power of a power rule is a fundamental property of exponents. In simple terms, it states that when you raise an exponent to another exponent, you separate the exponents by multiplication. The rule can be written as \[ (a^m)^n = a^{m \cdot n} \]where:

• \(a\) represents the base.
• \(m\) and \(n\) are the exponents.

To apply this rule, focus on nesting exponents. Always multiply the exponents when you see parentheses enclosing a power raised to another power. In the given problem, \((5^3)^x\) was transformed into a single power: \(5^{3x}\). This multiplication step simplifies calculations and makes exponential equations easier to solve.

Solving for \(x\) involves isolating the variable on one side of the equation. This is achieved by using basic algebraic principles such as addition, subtraction, multiplication, or division. After applying the power of a power rule, the given equation simplifies to \[5^{3x} = 5^{2x-3}\]Since both sides share the same base (\(5\)), the exponents themselves must be equal. Hence, we equate the exponents:\[3x = 2x - 3\]To isolate \(x\), subtract \(2x\) from both sides:

• \(3x - 2x = x\)
• This gives us \(x = -3\)

Each step moves us closer to a clear value, simplifying the equation step by step.

Rewriting bases in exponential equations helps in matching the powers conveniently. This process involves expressing numbers as powers of a common base. In the exercise, the numbers \(125\) and \(5\) needed to be compared. Recognizing the common base is crucial: both numbers can be written as powers of \(5\).

• \(125 = 5^3\)

Converting numbers to the same base facilitates easier equation manipulation since the exponents can be directly compared once the bases match. This strategic conversion is vital in solving exponential equations effectively. Once bases are harmonized, other properties of exponents, such as the power of a power rule, can be applied seamlessly, aiding in the resolution of the equation.