select all the expressions that are equivalent to ((6^2)^6). \n(6^7 cdot 6^5) (\frac{1}{6^8}) (3^{12} cdot…

Explanation:

First, calculate the value of ((6^{2})^{6}) using the power - of - a - power rule ((a^{m})^{n}=a^{mn}). So, ((6^{2})^{6}=6^{2\times6}=6^{12}).

Step 1: Analyze (6^{7}\cdot6^{5})

Using the product - of - powers rule (a^{m}\cdot a^{n}=a^{m + n}), for (6^{7}\cdot6^{5}), we have (m = 7) and (n = 5). Then (6^{7}\cdot6^{5}=6^{7 + 5}=6^{12}), which is equivalent to ((6^{2})^{6}).

Step 2: Analyze (\frac{1}{6^{8}})

Using the negative - exponent rule (a^{-n}=\frac{1}{a^{n}}), (\frac{1}{6^{8}}=6^{-8}), and (6^{-8}\neq6^{12}), so (\frac{1}{6^{8}}) is not equivalent to ((6^{2})^{6}).

Step 3: Analyze (3^{12}\cdot2^{12})

Using the product - of - powers rule for a product raised to a power ((ab)^{n}=a^{n}b^{n}), we know that (6 = 3\times2), so (6^{12}=(3\times2)^{12}=3^{12}\times2^{12}) (by ((ab)^{n}=a^{n}b^{n}) with (a = 3), (b = 2), and (n = 12)). So (3^{12}\cdot2^{12}) is equivalent to ((6^{2})^{6}).

Step 4: Analyze (6^{3}\cdot6^{4})

Using the product - of - powers rule (a^{m}\cdot a^{n}=a^{m + n}), for (6^{3}\cdot6^{4}), we have (m = 3) and (n = 4). Then (6^{3}\cdot6^{4}=6^{3+4}=6^{7}\neq6^{12}), so (6^{3}\cdot6^{4}) is not equivalent to ((6^{2})^{6}).

Answer:

The expressions equivalent to ((6^{2})^{6}) are (6^{7}\cdot6^{5}) and (3^{12}\cdot2^{12}) (i.e., the first and the third expressions in the given options).