Question

rewrite the following in fractional exponential form. (in alphabetical order \xyz\)\\(sqrt3{27x^{4}y^{2}z})

Explanation:

Step1: Recall the radical to exponential rule

The formula for converting a radical to a fractional exponent is $\sqrt[n]{a^m}=a^{\frac{m}{n}}$. Also, for a product inside a radical, we can use the property $\sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}$. So we can apply these rules to each factor in $\sqrt[3]{27x^{4}y^{2}z}$.

First, handle the constant term: $\sqrt[3]{27}$. Since $27 = 3^3$, then $\sqrt[3]{3^3}=3^{\frac{3}{3}} = 3^1=3$.

Step2: Handle the $x$ term

For the $x$ term, we have $\sqrt[3]{x^{4}}$. Using the rule $\sqrt[n]{a^m}=a^{\frac{m}{n}}$, here $n = 3$ and $m = 4$, so $\sqrt[3]{x^{4}}=x^{\frac{4}{3}}$.

Step3: Handle the $y$ term

For the $y$ term, $\sqrt[3]{y^{2}}$. Using the same rule, with $n = 3$ and $m = 2$, we get $\sqrt[3]{y^{2}}=y^{\frac{2}{3}}$.

Step4: Handle the $z$ term

For the $z$ term, $\sqrt[3]{z}=\ z^{\frac{1}{3}}$ (since $z = z^1$ and $n = 3$, $m = 1$).

Step5: Combine all the terms

Now, multiply all the simplified terms together: $3\times x^{\frac{4}{3}}\times y^{\frac{2}{3}}\times z^{\frac{1}{3}}$. In alphabetical order (xyz), this is $3x^{\frac{4}{3}}y^{\frac{2}{3}}z^{\frac{1}{3}}$.

Answer:

$3x^{\frac{4}{3}}y^{\frac{2}{3}}z^{\frac{1}{3}}$