Question

for the partial solution shown to the right, identify the property of exponents that is used.

the property is used in the partial solution.

\\(9^{\frac{x}{7} + 5} = 27^{\frac{x}{5}}\\)
\\((3^2)^{\frac{x}{7} + 5} = (3^3)^{\frac{x}{5}}\\)
\\(3^{\frac{2x}{7} + 10} = 3^{\frac{3x}{5}}\\)

Explanation:

Step1: Recall Exponent Properties

The general property for raising a power to a power is \((a^m)^n = a^{m \times n}\), where \(a\) is the base, \(m\) and \(n\) are exponents.

Step2: Analyze the Given Steps

In the step where \(9^{\frac{x}{7}+5}\) is rewritten as \((3^2)^{\frac{x}{7}+5}\) and \(27^{\frac{x}{5}}\) is rewritten as \((3^3)^{\frac{x}{5}}\), and then further simplified to \(3^{\frac{2x}{7}+10}\) and \(3^{\frac{3x}{5}}\) respectively, we apply the power - of - a - power property. Let's take the first transformation: for \((3^2)^{\frac{x}{7}+5}\), using the formula \((a^m)^n=a^{m\times n}\), here \(a = 3\), \(m=2\), \(n=\frac{x}{7}+5\), so \((3^2)^{\frac{x}{7}+5}=3^{2\times(\frac{x}{7}+5)}=3^{\frac{2x}{7}+10}\). Similarly, for \((3^3)^{\frac{x}{5}}\), using the same property, \(a = 3\), \(m = 3\), \(n=\frac{x}{5}\), so \((3^3)^{\frac{x}{5}}=3^{3\times\frac{x}{5}}=3^{\frac{3x}{5}}\).

Answer:

Power - of - a - Power (or \((a^m)^n=a^{mn}\))