select all equivalent expressions.\n\\(\\left(5^{3} \\times 5^{(-2)}\\right)^{4}\\)\n\\(5^{12} \\times…

Explanation:

Step1: Simplify the exponent inside the parentheses

First, use the rule of exponents (a^m \times a^n = a^{m + n}) for the expression inside the parentheses (5^3 \times 5^{-2}). So we have (5^{3+(-2)} = 5^{1}).

Step2: Apply the power of a power rule

Now, apply the power of a power rule ((a^m)^n=a^{m\times n}) to ((5^{1})^4). This gives us (5^{1\times4}=5^4).

Step3: Analyze the first option

For the first option (5^{12}\times5^{-8}), use the rule (a^m\times a^n = a^{m + n}), so (5^{12+(-8)}=5^{4}), which is equivalent to our simplified expression. Also, ((5^3\times5^{-2})^4=(5^{3\times4})\times(5^{-2\times4}) = 5^{12}\times5^{-8}) by the power of a product rule ((ab)^n=a^n b^n).

Step4: Calculate the numerical value of (5^4)

Calculate (5^4 = 5\times5\times5\times5=625), so the third option (625) is equivalent.

Step5: Analyze the last option

The last option is (\frac{5^{-12}}{5^{8}}), using the rule (\frac{a^m}{a^n}=a^{m - n}), we get (5^{-12 - 8}=5^{-20}), which is not equivalent. The second option (\frac{1}{625}=5^{-4}), not equivalent. The fifth option (5^{-24}) is not equivalent.

Answer:

A. (5^{12} \times 5^{-8}), C. (625), D. (5^{4})