Answer
a. $\sqrt[4] {27}$
b. $\sqrt[6] {x^5}$
c. $\sqrt[6] {16,807}$
Work Step by Step
a. Convert radicals to exponential expressions:
$3^{\frac{1}{2}} \cdot 3^{\frac{1}{4}}$
When two exponential expressions with the same base are multiplied together, add the exponents together, keeping the base as-is:
$=3^{\frac{1}{2} + \frac{1}{4}}$
Convert the fractions in the exponents so that they both have the same denominator:
$=3^{\frac{2}{4} + \frac{1}{4}}$
$=3^{\frac{3}{4}}$
Convert to radical form:
$=\sqrt[4] {27}$
b. Convert radicals to exponential expressions:
$\dfrac{x^{\frac{3}{2}}}{x^{\frac{2}{3}}}$
When one exponential expression is divided by another exponential expression with the same base, subtract the exponents, keeping the base as-is:
$=x^{\frac{3}{2} - \frac{2}{3}}$
Convert the fractions in the exponents so that they both have the same denominator:
$=x^{\frac{9}{6} - \frac{4}{6}}$
Subtract the exponents:
$=x^{\frac{5}{6}}$
Convert to radical form:
$=\sqrt[6] {x^5}$
c. Convert radicals to exponential expressions:
$7^{\frac{1}{2}} \cdot 7^{\frac{1}{3}}$
When two exponential expressions with the same base are multiplied together, add the exponents together, keeping the base as-is:
$=7^{\frac{1}{2} + \frac{1}{3}}$
Convert the fractions in the exponents so that they both have the same denominator:
$=7^{\frac{3}{6} + \frac{2}{6}}$
Add the exponents:
$=7^{\frac{5}{6}}$
Convert to radical form:
$=\sqrt[6] {16,807}$
