To determine whether the expression [tex]\( x^3 \cdot x^3 \cdot x^3 \)[/tex] is equivalent to [tex]\( x^3 \cdot 3^3 \)[/tex], we need to analyze and simplify both expressions carefully.
### Simplifying [tex]\( x^3 \cdot x^3 \cdot x^3 \)[/tex]
First, let's simplify [tex]\( x^3 \cdot x^3 \cdot x^3 \)[/tex]. This is a product of the same base ([tex]\(x\)[/tex]) raised to the power of 3 multiple times.
Using the properties of exponents, specifically the product of powers property which states [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex], we can combine the exponents:
[tex]\[ x^3 \cdot x^3 \cdot x^3 = x^{3+3+3} = x^9 \][/tex]
Therefore, the simplified form of [tex]\( x^3 \cdot x^3 \cdot x^3 \)[/tex] is [tex]\( x^9 \)[/tex].
### Analyzing [tex]\( x^3 \cdot 3^3 \)[/tex]
Now, let's look at [tex]\( x^3 \cdot 3^3 \)[/tex]. This expression involves two different bases, [tex]\(x\)[/tex] and [tex]\(3\)[/tex], both raised to the power of 3. Since the bases are different, we cannot combine the terms using exponent rules. Thus, [tex]\( x^3 \cdot 3^3 \)[/tex] cannot be simplified further and remains as it is.
### Comparing the Expressions
Let's now compare the simplified form of the first expression with the second expression:
- The simplified form of [tex]\( x^3 \cdot x^3 \cdot x^3 \)[/tex] is [tex]\( x^9 \)[/tex].
- The second expression [tex]\( x^3 \cdot 3^3 \)[/tex] remains [tex]\( x^3 \cdot 3^3 \)[/tex] because the bases are different.
Since [tex]\( x^9 \)[/tex] and [tex]\( x^3 \cdot 3^3 \)[/tex] are not the same, it is clear that:
[tex]\[ x^3 \cdot x^3 \cdot x^3 \neq x^3 \cdot 3^3 \][/tex]
### Conclusion
The expression [tex]\( x^3 \cdot x^3 \cdot x^3 \)[/tex] is not equivalent to [tex]\( x^3 \cdot 3^3 \)[/tex]. The former simplifies to [tex]\( x^9 \)[/tex], while the latter cannot be simplified further and remains as [tex]\( x^3 \cdot 3^3 \)[/tex]. The difference in bases and the resultant forms make the two expressions distinct.
