To correctly develop expressions using the power rule of exponents, we can apply the following rules:
- The power of a product rule: \((a^m)(a^n) = a^{m+n}\)
- The power of a power rule: \((a^m)^n = a^{m \cdot n}\)
- The zero exponent rule: \(a^0 = 1\) (for any \(a \neq 0\))
Let's look at each expression:
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\(12^9\): This expression is already simplified and doesn't indicate further simplification using the power rules.
-
\((-5)^0\): By the zero exponent rule, this simplifies to 1.
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\((7^8) \cdot (7^{10})\): Using the power of a product rule, this simplifies to:
\((7^8) \cdot (7^{10}) = 7^{8+10} = 7^{18}\). -
\((7^4)^8\): By the power of a power rule, this simplifies to:
\((7^4)^8 = 7^{4 \cdot 8} = 7^{32}\).
So, the expressions that are developed using the power rule of exponents are:
- \((7^8) \cdot (7^{10})\) which simplifies to \(7^{18}\).
- \((7^4)^8\) which simplifies to \(7^{32}\).
If the question is asking which expressions can be further simplified using the power rule, then options 3 and 4 are the ones that apply.