Bot GPT 4-o mini To determine which expression is correctly developed using the Product Rule of Exponents, we need to understand the Product Rule itself. The Product Rule states that when multiplying two numbers with the same base, you add the exponents:
\[ a^m \cdot a^n = a^{m+n} \]
Let's analyze each response to see if it adheres to this rule.
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327: This is not an expression involving exponents; it's simply a number.
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32: Like the first response, this is just a number, not an expression using exponents in the product form.
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7: This is just a number, not an expression involving exponents.
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32 superscript 7 baseline: This is the expression \( 3^2 \cdot 3^7 \). By the Product Rule, \( 3^2 \cdot 3^7 = 3^{2+7} = 3^9 \). This is a correctly developed expression using the Product Rule.
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62⋅73: This can be written as \( 6^2 \cdot 7^3 \). Since these bases are different, the Product Rule does not apply directly in the context of combining these exponents.
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10^8 ⋅ 10^8: This expression combines two of the same base, 10. The exponents can be added, so \( 10^8 \cdot 10^8 = 10^{8+8} = 10^{16} \). This expression also correctly uses the Product Rule.
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(52)9: This can be interpreted as \( (5^2)^9 \), which would apply the power of a power rule rather than the product rule.
So, the correctly developed expression that uses the Product Rule of Exponents is 32 superscript 7 baseline (3^2 ⋅ 3^7) and 10^8 ⋅ 10^8 which simplifies down to \( 10^{16} \).
1 answer