To simplify the expression [tex]\( x^5 \cdot x^7 \)[/tex], we need to apply the properties of exponents. Specifically, there's a rule that tells us how to handle the multiplication of powers with the same base.
When you multiply two expressions with the same base, you add the exponents. In general,
[tex]\[ a^m \cdot a^n = a^{m+n} \][/tex]
In this case, our base is [tex]\( x \)[/tex], and our exponents are [tex]\( 5 \)[/tex] and [tex]\( 7 \)[/tex]. Therefore, we add the exponents:
[tex]\[ x^5 \cdot x^7 = x^{5+7} \][/tex]
Now, compute the sum of the exponents:
[tex]\[ 5 + 7 = 12 \][/tex]
Thus, the simplified form of the expression is:
[tex]\[ x^{12} \][/tex]
So, the correct answer is:
[tex]\[ x^{12} \][/tex]
