Question

Convert the following expressions with rational
exponents to the radical form
(6^2/3)^1/2

Answer 1

Answer:

$\sqrt[3]{6}$

Step-by-step explanation:

To convert the expression $(6^{\frac{2}{3}})^{\frac{1}{2}}$ from rational exponent form to radical form, we can use the property that $(a^b)^c = a^{b \cdot c}$. Applying this property, we can simplify the given expression:

$(6^{\frac{2}{3}})^{\frac{1}{2}} = 6^{\frac{2}{3} \cdot \frac{1}{2}}$

Now, multiply the exponents:

$6^{\frac{2}{6}}$

The exponent $\frac{2}{6}$ can be simplified to $\frac{1}{3}$. Therefore, the expression becomes:

$6^{\frac{1}{3}}$

Now, $6^{\frac{1}{3}}$ represents the cube root of 6. So, in radical form:

$(6^{\frac{2}{3}})^{\frac{1}{2}} = \sqrt[3]{6}$

Thus, the expression $(6^{\frac{2}{3}})^{\frac{1}{2}}$ in radical form is:

$\sqrt[3]{6}$

Answer 2

Answer:

$\large\text{ \sqrt[3]{6} }$

Step-by-step explanation:

A radical represents a root of a number and can appear in different forms, including square root, cube root, fourth root, etc. The index of the radical specifies which root is being taken.

To convert the given expression to its radical form, we can begin by using the power of a power exponent rule.

$\boxed{\begin{array}{c}\underline{\textsf{Power of a Power Exponent Rule}}\\\\\Large\text{ \left(a^b\right)^c=a^{bc} }\end{array}}$

In this case, 2/3 is the first exponent and 1/2 is the second exponent. To apply the exponent rule, multiply the exponents:

$\large\text{ \left(6^{\frac23}\right)^{\frac12}=6^{\frac23\times \frac12}=6^{\frac13} }$

Now, use the fractional exponent rule:

$\boxed{\begin{array}{c}\underline{\textsf{Fractional Exponent Rule}}\\\\\Large\text{ a^{\frac{m}{n}}=\sqrt[n]{a^m} }\end{array}}$

In this case, m = 1 and n = 3, so:

$\large\text{ \sqrt[3]{6^1}=\sqrt[3]{6} }$

So the given expression converted to its radical form is:

$\Large\boxed{\boxed{\sqrt[3]{6}}}$