To determine whether the expression [tex]\(x^3 \cdot x^3 \cdot x^3\)[/tex] is equivalent to [tex]\(x^3 \cdot 3 \cdot 3\)[/tex], let's break down and simplify each expression step by step.
### Simplifying [tex]\(x^3 \cdot x^3 \cdot x^3\)[/tex]:
1. Combine the Exponents:
When multiplying powers with the same base, you add the exponents. Thus:
[tex]\[
x^3 \cdot x^3 = x^{3+3} = x^6
\][/tex]
2. Further Multiplying:
Now multiply [tex]\(x^6\)[/tex] by another [tex]\(x^3\)[/tex]:
[tex]\[
x^6 \cdot x^3 = x^{6+3} = x^9
\][/tex]
Therefore, the expression [tex]\(x^3 \cdot x^3 \cdot x^3\)[/tex] simplifies to:
[tex]\[
x^9
\][/tex]
### Simplifying [tex]\(x^3 \cdot 3 \cdot 3\)[/tex]:
1. Combine the Constants:
Multiply the numerical coefficients first:
[tex]\[
3 \cdot 3 = 9
\][/tex]
2. Include the Variable:
Now, multiply the result by [tex]\(x^3\)[/tex]:
[tex]\[
x^3 \cdot 9 = 9x^3
\][/tex]
Therefore, the expression [tex]\(x^3 \cdot 3 \cdot 3\)[/tex] simplifies to:
[tex]\[
9x^3
\][/tex]
### Comparing the Results:
- [tex]\(x^3 \cdot x^3 \cdot x^3\)[/tex] simplifies to [tex]\(x^9\)[/tex]
- [tex]\(x^3 \cdot 3 \cdot 3\)[/tex] simplifies to [tex]\(9x^3\)[/tex]
These simplified expressions, [tex]\(x^9\)[/tex] and [tex]\(9x^3\)[/tex], are not equivalent because they represent different mathematical quantities. One represents [tex]\(x\)[/tex] raised to the ninth power, and the other represents nine times [tex]\(x\)[/tex] cubed.
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.
