To determine which expressions are equivalent to [tex]\( 4^3 \cdot 4^x \)[/tex], let's start by simplifying the original expression.
Given:
[tex]\[ 4^3 \cdot 4^x \][/tex]
Using the property of exponents that states [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex], we can combine the exponents:
[tex]\[ 4^3 \cdot 4^x = 4^{3+x} \][/tex]
So, the simplified form of the original expression is [tex]\( 4^{3+x} \)[/tex].
Now, let's check each given expression to see if it matches [tex]\( 4^{3+x} \)[/tex].
A. [tex]\( 4^{3-x} \)[/tex]:
This expression does not match [tex]\( 4^{3+x} \)[/tex]. Thus, it is not equivalent.
B. [tex]\( 4^{3+x} \)[/tex]:
This matches the simplified form exactly. Thus, it is equivalent.
C. [tex]\( (4 \cdot x)^3 \)[/tex]:
Simplifying this expression:
[tex]\[ (4 \cdot x)^3 = 4^3 \cdot x^3 \][/tex]
This does not match [tex]\( 4^{3+x} \)[/tex]. Thus, it is not equivalent.
D. [tex]\( 64 \cdot 4^x \)[/tex]:
Simplify [tex]\( 64\)[/tex]:
[tex]\[ 64 = 4^3 \][/tex]
Rewrite the expression using this fact:
[tex]\[ 64 \cdot 4^x = 4^3 \cdot 4^x \][/tex]
Which simplifies to:
[tex]\[ 4^{3+x} \][/tex]
This matches the simplified form exactly. Thus, it is equivalent.
E. [tex]\( 16^{3 x} \)[/tex]:
Rewrite 16 as [tex]\( 4^2 \)[/tex]:
[tex]\[ 16 = 4^2 \][/tex]
So:
[tex]\[ 16^{3 x} = (4^2)^{3 x} \][/tex]
Using the power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ (4^2)^{3 x} = 4^{2 \cdot 3 x} = 4^{6 x} \][/tex]
This does not match [tex]\( 4^{3+x} \)[/tex]. Thus, it is not equivalent.
F. [tex]\( 4^{3 x} \)[/tex]:
This does not match [tex]\( 4^{3+x} \)[/tex]. Thus, it is not equivalent.
So, the expressions that are equivalent to [tex]\( 4^3 \cdot 4^x \)[/tex] are:
B. [tex]\( 4^{3+x} \)[/tex]
D. [tex]\( 64 \cdot 4^x \)[/tex]
