Question

select all equivalent expressions.
\\(\left(5^{3} \times 5^{(-2)}\
ight)^{4}\\)
\\(5^{12} \times 5^{(-8)}\\)
\\(\frac{1}{625}\\)
\\(625\\)
\\(5^{4}\\)
\\(5^{-24}\\)
\\(\frac{5^{(-12)}}{5^{8}}\\)

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. \(5^{4}\)