Question
Why can’t you use the product of powers rule to simplify this expression? Explain.
34 · 28
Answers
GPT-4o mini
The product of powers rule states that when you multiply two expressions that have the same base, you can add their exponents. This rule can be stated mathematically as:
\[
a^m \cdot a^n = a^{m+n}
\]
However, in the expression \(34 \cdot 28\), the bases are not the same. The base of the first term (34) is different from the base of the second term (28). Therefore, you cannot apply the product of powers rule because it specifically applies only to terms with identical bases.
To further clarify:
1. The numbers 34 and 28 are not in exponential form with the same base.
2. \(34\) can be rewritten as \(2 \cdot 17\), and \(28\) can be rewritten as \(4 \cdot 7\); however, they still do not share common bases in a form that fits the product of powers rule.
Thus, since the bases of the numbers in the expression are different, you cannot use the product of powers rule for simplification. Instead, you can multiply the numbers directly or use other algebraic methods, but not the product of powers rule.
\[
a^m \cdot a^n = a^{m+n}
\]
However, in the expression \(34 \cdot 28\), the bases are not the same. The base of the first term (34) is different from the base of the second term (28). Therefore, you cannot apply the product of powers rule because it specifically applies only to terms with identical bases.
To further clarify:
1. The numbers 34 and 28 are not in exponential form with the same base.
2. \(34\) can be rewritten as \(2 \cdot 17\), and \(28\) can be rewritten as \(4 \cdot 7\); however, they still do not share common bases in a form that fits the product of powers rule.
Thus, since the bases of the numbers in the expression are different, you cannot use the product of powers rule for simplification. Instead, you can multiply the numbers directly or use other algebraic methods, but not the product of powers rule.