Sure, let's walk through the step-by-step process to fully simplify Francisco and Ryan's expressions and then determine if they started with equivalent expressions.
### Francisco's Expression
Francisco's original expression is:
[tex]\[
\frac{x^{\frac{1}{2}}}{\frac{3}{18}}
\][/tex]
1. Simplify the denominator:
[tex]\[
\frac{3}{18} = \frac{1}{6}
\][/tex]
2. Rewrite the expression:
[tex]\[
\frac{x^{\frac{1}{2}}}{\frac{1}{6}}
\][/tex]
3. Dividing by a fraction is equivalent to multiplying by its reciprocal:
[tex]\[
x^{\frac{1}{2}} \times 6 = 6x^{\frac{1}{2}}
\][/tex]
Thus, the simplified expression for Francisco is:
[tex]\[
6x^{\frac{1}{2}}
\][/tex]
### Ryan's Expression
Ryan's original expression is:
[tex]\[
\sqrt[27]{x^2 \cdot x^3 \cdot x^4}
\][/tex]
1. Combine the exponents inside the radical:
[tex]\[
x^2 \cdot x^3 \cdot x^4 = x^{2+3+4} = x^9
\][/tex]
2. Apply the radical:
[tex]\[
\sqrt[27]{x^9} = (x^9)^{\frac{1}{27}}
\][/tex]
3. Simplify the exponent:
[tex]\[
(x^9)^{\frac{1}{27}} = x^{\frac{9}{27}} = x^{\frac{1}{3}}
\][/tex]
Thus, Ryan’s simplified expression is:
[tex]\[
x^{\frac{1}{3}}
\][/tex]
### Comparison of Francisco’s and Ryan’s Expressions
Francisco’s simplified expression is:
[tex]\[
6x^{\frac{1}{2}}
\][/tex]
Ryan’s simplified expression is:
[tex]\[
x^{\frac{1}{3}}
\][/tex]
Clearly, the simplified expressions [tex]\(6x^{\frac{1}{2}}\)[/tex] and [tex]\(x^{\frac{1}{3}}\)[/tex] are not equivalent. Francisco and Ryan did not start with equivalent expressions because the simplified forms do not match.
### Conclusion
Francisco's original expression, when simplified correctly, results in [tex]\(6x^{\frac{1}{2}}\)[/tex], while Ryan's original expression results in [tex]\(x^{\frac{1}{3}}\)[/tex]. Therefore, Francisco and Ryan did not start with expressions that are equal; rather, their initial expressions are fundamentally different and simplify to non-equivalent results.
