Let's carefully analyze the given expressions to determine if they are equivalent.
1. First Expression:
[tex]\[
x^3 \cdot x^3 \cdot x^3
\][/tex]
Step-by-step, we can simplify this using the laws of exponents. When multiplying powers with the same base, we add the exponents:
[tex]\[
x^3 \cdot x^3 \cdot x^3 = x^{3+3+3} = x^9
\][/tex]
So, the first expression simplifies to [tex]\( x^9 \)[/tex].
2. Second Expression:
[tex]\[
x^3 \cdot 3 \cdot 3
\][/tex]
Here, we first combine the constants:
[tex]\[
3 \cdot 3 = 9
\][/tex]
Thus, the expression becomes:
[tex]\[
x^3 \cdot 9
\][/tex]
This can be written as:
[tex]\[
9x^3
\][/tex]
So, the second expression simplifies to [tex]\( 9x^3 \)[/tex].
Now, let’s compare the simplified forms of both expressions:
- The first expression simplifies to [tex]\( x^9 \)[/tex].
- The second expression simplifies to [tex]\( 9x^3 \)[/tex].
Clearly, [tex]\( x^9 \)[/tex] and [tex]\( 9x^3 \)[/tex] are not the same.
Therefore, the expressions [tex]\( x^3 \cdot x^3 \cdot x^3 \)[/tex] and [tex]\( x^3 \cdot 3 \cdot 3 \)[/tex] are not equivalent. This conclusion holds because their simplified forms differ, with [tex]\( x^9 \)[/tex] and [tex]\( 9x^3 \)[/tex] respectively.
