Simplify. Express answers in rational form with positive exponents.

Hi emilyliv

The best tip to give you is the follow the order of operations: BEDMAS.

Start from the innermost brackets first:

\(\left(\frac{\left(x^{18}\right)^{\frac{-1}{6}}}{\sqrt[5]{243x^{10}}}\right)^{0.5}\)

Note that x^18 cannot be reduce further, so multiply the -1/6 into it; by doing that 18(-1/6) is -3.

\(\left(\frac{x^{-3}}{\sqrt[5]{243x^{10}}}\right)^{0.5}\)

Now, at the bottom, we have the 5th root of 243x^10. Feel free to make the fifth root into an exponent of 1/5:

\(\left(\frac{x^{-3}}{\left(243x^{10}\right)^{\frac{1}{5}}}\right)^{0.5}\)

Now distribute that 1/5 into the factors 243 and x^10 to get:

\(\left(\frac{x^{-3}}{3x^2}\right)^{0.5}\)

Now, simplify the fraction further using the quotient rule:

\(\left(\frac{1}{3}x^{-5}\right)^{0.5}\)

Now we cannot simplify what's into the brackets further, we distribute the outer exponent of 0.5 to all the factors: 1/3 and x^-5:

\(\left(\frac{1}{3}\right)^{0.5}\left(x^{-5}\right)^{0.5}\)

Simplify further:

\(\left(\frac{1}{\sqrt{3}}\right)\left(x^{-\frac{5}{2}}\right)\)

Simplify more:

\(\left(\frac{1}{\sqrt{3}}\right)\left(\frac{1}{x^{\frac{5}{2}}}\right)\)

More:

\(\left(\frac{1}{\sqrt{3}}\right)\left(\frac{1}{\sqrt{x^5}}\right)\)

More:

\(\frac{1}{\sqrt{3}\cdot \sqrt{x^5}}\)

More if you like:

\(\frac{1}{\sqrt{3x^5}}\)

More if you like:

\(\frac{1}{x^2\sqrt{3x}}\)

Finally, you want to "rationalize", meaning no square-root in the denominator:

\(\frac{\sqrt{3x}}{3x^3}\)

Let me know if you have more questions on this topic or on this question, GL

okay, that makes a lot more sense, thank u!