Sure, let's go through the problem step by step:
1. Understanding the rule for multiplying exponents: When you multiply two exponential terms that have the same base, you add the exponents. This rule can be stated as:
[tex]\[
a^m \times a^n = a^{m+n}
\][/tex]
2. Applying the rule to specific examples: Let's consider specific values for the base [tex]\(a\)[/tex] and the exponents [tex]\(m\)[/tex] and [tex]\(n\)[/tex]. For example, let [tex]\(a = 2\)[/tex], [tex]\(m = 3\)[/tex], and [tex]\(n = 4\)[/tex].
3. Adding the exponents: According to the rule, we add the exponents [tex]\(m\)[/tex] and [tex]\(n\)[/tex]:
[tex]\[
m + n = 3 + 4 = 7
\][/tex]
4. Raising the base to the new exponent: Now, we raise the base [tex]\(a\)[/tex] to the power of the new exponent we just calculated:
[tex]\[
a^{m+n} = 2^7
\][/tex]
5. Calculating the result: Finally, we calculate the value of [tex]\(2^7\)[/tex]:
[tex]\[
2^7 = 128
\][/tex]
Therefore, combining all the steps together, we have:
[tex]\[
2^3 \times 2^4 = 2^{3+4} = 2^7 = 128
\][/tex]
So, the result of [tex]\(a^m \times a^n\)[/tex] for [tex]\(a = 2\)[/tex], [tex]\(m = 3\)[/tex], and [tex]\(n = 4\)[/tex] is:
[tex]\[
2^3 \times 2^4 = 2^7 = 128
\][/tex]
The combined result step is:
[tex]\[
(7, 128)
\][/tex]
In general terms, if we denote the base as [tex]\(a\)[/tex], and the exponents as [tex]\(m\)[/tex] and [tex]\(n\)[/tex], the final result of [tex]\(a^m \times a^n\)[/tex] is:
[tex]\[
a^{m+n}
\][/tex]
For the specific numbers provided:
[tex]\[
a^3 \times a^4 = a^{7} = 128 \][/tex]
Thus yielding the results [tex]\((7, 128)\)[/tex].
