Answer:
Yes, they started with equal expressions.
Step-by-step explanation:
To simplify Vanessa's expression, we must rationalize the denominator. This means we must make the denominator x. We have
[tex]\frac{x^{\frac{4}{3}}}{x^{\frac{5}{6}}}[/tex]
We need to multiply the denominator by x^(1/6) to make a whole x; this means we must multiply the numerator by x^(1/6) as well:
[tex]\frac{x^{\frac{4}{3}}}{x^{\frac{5}{6}}} \times \frac{x^{\frac{1}{6}}}{x^{\frac{1}{6}}}}\\\\=\frac{x^{\frac{4}{3}+\frac{1}{6}}}{x^{\frac{5}{6}+\frac{1}{6}}}\\\\=\frac{x^{\frac{8}{6}+\frac{1}{6}}}{x^\frac{6}{6}}}\\\\=\frac{x^{\frac{9}{6}}}{x}[/tex]
Now we can separate the x^(9/6) in the numerator:
[tex]\frac{x^{\frac{9}{6}}}{x}\\\\=\frac{x^{\frac{6}{6}+\frac{3}{6}}}{x}\\\\=\frac{x^{\frac{6}{6}}(x^{\frac{3}{6}})}{x}\\\\=\frac{x(x^{\frac{3}{6}})}{x}\\\\=x^{\frac{3}{6}}=x^{\frac{1}{2}}=\sqrt{x}[/tex]
William's expression is
[tex](x\times x^3 \times x^4)^{\frac{1}{16}}\\[/tex]
We first use the product property to simplify inside parentheses:
[tex](x^{1+3+4})^{\frac{1}{16}}\\\\(x^8)^{\frac{1}{16}}[/tex]
Now we use the product of a product property to multiply the exponents:
[tex](x^8)^{\frac{1}{16}}\\\\=x^{\frac{8}{16}}\\\\=x^{\frac{1}{2}}=\sqrt{x}[/tex]
Therefore they started with equal expressions.
