Question

which two expressions are equivalent to each other?
\\(\frac{4^{20}}{4^{-4}}\\)
\\(\left(4^{-2}\
ight)^{-8}\\)
\\(4^{10}\cdot 4^{6}\\)
\\(4^{10}\cdot 4^{-6}\\)

Explanation:

Step1: Simplify \(\frac{4^{20}}{4^{-4}}\)

Using the rule \( \frac{a^m}{a^n}=a^{m - n} \), we have \( \frac{4^{20}}{4^{-4}}=4^{20-(-4)} = 4^{24} \).

Step2: Simplify \((4^{-2})^{-8}\)

Using the rule \( (a^m)^n=a^{m\times n} \), we get \( (4^{-2})^{-8}=4^{(-2)\times(-8)} = 4^{16} \).

Step3: Simplify \(4^{10}\cdot4^{6}\)

Using the rule \( a^m\cdot a^n=a^{m + n} \), we have \( 4^{10}\cdot4^{6}=4^{10 + 6}=4^{16} \).

Step4: Simplify \(4^{10}\cdot4^{-6}\)

Using the rule \( a^m\cdot a^n=a^{m + n} \), we get \( 4^{10}\cdot4^{-6}=4^{10+(-6)} = 4^{4} \).

From the above simplifications, we can see that \((4^{-2})^{-8}\) and \(4^{10}\cdot4^{6}\) both simplify to \(4^{16}\), so they are equivalent.

Answer:

\(\boldsymbol{(4^{-2})^{-8}}\) and \(\boldsymbol{4^{10}\cdot4^{6}}\)