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Which expression is correctly developed to use the power rule of exponents and parentheses seven with eight exponent times in parentheses seven with a 10 exponent or would it be 12 with a nine exponent or in parentheses seven with a four exponent and then an eight exponent or would it be in parentheses -5 and an exponent of zero

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To properly apply the power rule of exponents and simplify the expressions you've provided, let's clarify the expressions and their exponents. The power rule states that \((a^m)^n = a^{m \cdot n}\).

Let's break down the possible options you've mentioned:

  1. \( (7^8)^{10} \): Using the power rule, this would be simplified to \( 7^{8 \cdot 10} = 7^{80} \).

  2. \( (7^12)^{9} \): Using the power rule, this would be simplified to \( 7^{12 \cdot 9} = 7^{108} \).

  3. \( (7^4)^{8} \): Using the power rule, this would be simplified to \( 7^{4 \cdot 8} = 7^{32} \).

  4. \( (-5)^0 \): Any non-zero expression raised to the power of zero equals 1, so \( (-5)^0 = 1 \).

Thus, the expressions that can be developed using the power rule of exponents are:

  • \(7^8\) raised to the 10th power simplifies to \(7^{80}\).
  • \(7^{12}\) raised to the 9th power simplifies to \(7^{108}\).
  • \(7^{4}\) raised to the 8th power simplifies to \(7^{32}\).

The expression \((-5)^0\) does not require exponent rules in the traditional sense as it simply evaluates to 1 due to the zero exponent rule.

If this isn't what you were looking for, please clarify your question, and I'll be happy to assist you further!