Sure, let's analyze the given mathematical expression step by step.
We have the expression:
[tex]\[ 4^3 \cdot 4^x \][/tex]
According to the properties of exponents, when multiplying two expressions with the same base, we add their exponents. That is, for any base \( a \) and exponents \( m \) and \( n \):
[tex]\[ a^m \cdot a^n = a^{m+n} \][/tex]
Applying this property to our expression:
[tex]\[ 4^3 \cdot 4^x = 4^{3+x} \][/tex]
So, the correct answer is \( 4^{3+x} \).
Let's verify this against the provided options:
A. \( 4^{3x} \) – This is not correct.
B. \( (4 \cdot x)^3 \) – This is not correct.
C. \( 4^{3-x} \) – This is not correct.
D. \( 64 \cdot 4^x \) – This is not correct. While \( 4^3 = 64 \), the expression manipulates the base incorrectly.
E. \( 4^{3+x} \) – This matches our result.
F. \( 16^{3x} \) – This is not correct.
Thus, the correct answer is:
E. [tex]\( 4^{3+x} \)[/tex]
