Question

is the equation $5^{-5} \cdot (\sqrt3{5})^{12} = \frac{1}{5}$ true or false? enter 1 for true or 2 for false. (1 point)

Explanation:

Step1: Simplify the radical term

First, recall that \(\sqrt[3]{5}=5^{\frac{1}{3}}\). So, \((\sqrt[3]{5})^{12}=(5^{\frac{1}{3}})^{12}\).
Using the exponent rule \((a^{m})^{n}=a^{mn}\), we get \((5^{\frac{1}{3}})^{12}=5^{\frac{12}{3}} = 5^{4}\).

Step2: Simplify the left - hand side of the equation

The left - hand side of the equation is \(5^{-5}\cdot(\sqrt[3]{5})^{12}\). Substituting the simplified form of \((\sqrt[3]{5})^{12}\) we found above, we have \(5^{-5}\cdot5^{4}\).
Using the exponent rule \(a^{m}\cdot a^{n}=a^{m + n}\), we get \(5^{-5 + 4}=5^{-1}\).
Since \(a^{-n}=\frac{1}{a^{n}}\), \(5^{-1}=\frac{1}{5}\).

Step3: Compare the left - hand side and the right - hand side

The left - hand side simplifies to \(\frac{1}{5}\) and the right - hand side is also \(\frac{1}{5}\). So, the equation \(5^{-5}\cdot(\sqrt[3]{5})^{12}=\frac{1}{5}\) is true.

Answer:

1