Simplify the following expression:

[tex]\[ 4^3 \cdot 4^x \][/tex]

A. \(4^{3 x}\)
B. \((4 \cdot x)^3\)
C. \(4^{3-x}\)
D. \(64 \cdot 4^x\)
E. \(4^{3+x}\)
F. [tex]\(16^{3 x}\)[/tex]



Answer :

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]