Question 5 (Essay Worth 2 points)

Is the expression [tex]$x^3 \cdot x^3 \cdot x^3$[/tex] equivalent to [tex]$x^3 \cdot 3^3$[/tex]? Why or why not? Explain your reasoning.

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[tex]Q \sim A I \sim A \equiv \sim \underline{A} \sim \underline{A} \simeq[/tex]

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Answer :

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.