select all the expressions that are equivalent to $(6^{5})^{7}$. $6^{35}$, $6^{6} cdot 6^{2}$…
select all the expressions that are equivalent to $(6^{5})^{7}$. $6^{35}$, $6^{6} cdot 6^{2}$, $(6^{7})^{5}$, $\frac{1}{6^{35}}$
Answer
Explanation:
Step1: Recall exponent rule ((a^m)^n = a^{m\times n})
For ((6^5)^7), apply the rule: (5\times7 = 35), so ((6^5)^7=6^{35}).
Step2: Analyze (6^6\cdot6^2)
Use exponent rule (a^m\cdot a^n=a^{m + n}). Here, (6+2 = 8), so (6^6\cdot6^2 = 6^8\neq6^{35}).
Step3: Analyze ((6^7)^5)
Apply ((a^m)^n=a^{m\times n}): (7\times5 = 35), so ((6^7)^5=6^{35}).
Step4: Analyze (\frac{1}{6^{35}})
(\frac{1}{6^{35}}=6^{-35}\neq6^{35}).
So the correct expressions are (6^{35}) and ((6^7)^5).
Answer:
A. (6^{35}), C. ((6^7)^5) (assuming the options are labeled as A: (6^{35}), B: (6^6\cdot6^2), C: ((6^7)^5), D: (\frac{1}{6^{35}}))