Let's examine the given expressions to determine if they are equivalent.
1. The first expression is [tex]\( x^3 \cdot x^3 \cdot x^3 \)[/tex]:
- When multiplying terms with the same base, you add their exponents. Hence, [tex]\( x^3 \cdot x^3 \cdot x^3 \)[/tex] can be simplified by adding the exponents: [tex]\( 3 + 3 + 3 \)[/tex].
- This results in [tex]\( x^{9} \)[/tex].
2. The second expression is [tex]\( x^{3 \cdot 3} \cdot 3 \)[/tex]:
- First, evaluate the exponent part [tex]\( 3 \cdot 3 \)[/tex], which equals 9.
- This results in [tex]\( x^{9} \cdot 3 \)[/tex].
Now we need to compare [tex]\( x^{9} \)[/tex] and [tex]\( x^{9} \cdot 3 \)[/tex]:
- [tex]\( x^{9} \)[/tex] is the simplified form of the first expression.
- [tex]\( x^{9} \cdot 3 \)[/tex] is the result of the second expression.
Clearly, these two expressions are not equivalent. The presence of the extra factor of 3 in the second expression [tex]\( x^{9} \cdot 3 \)[/tex] makes it larger than [tex]\( x^{9} \)[/tex] by a factor of 3.
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. The main difference is that [tex]\( x^{9} \)[/tex] is a simple power of [tex]\( x \)[/tex], while [tex]\( x^{9} \cdot 3 \)[/tex] multiplies that power by an additional factor of 3.
