Question

the quotient property of radicals requires the indices of the radicals to be the same. does this mean that it is not possible to write \\(\frac{\sqrt4{5}}{\sqrt{2}}\\) as a single radical? explain.

Explanation:

Analyze the given expression

Using the Quotient Property of Radicals and Rational Exponents knowledge points

$$ \frac{\sqrt[4]{x}}{\sqrt{x}} = \frac{x^{1/4}}{x^{1/2}} $$

Find a common denominator for the exponents

Using the Rational Exponents knowledge point

$$ \frac{1}{4} = \frac{1}{4}, \quad \frac{1}{2} = \frac{2}{4} $$

Rewrite with a common index

Using the Rational Exponents knowledge point

$$ \frac{x^{1/4}}{x^{2/4}} = \frac{\sqrt[4]{x}}{\sqrt[4]{x^2}} $$

Apply the quotient property

Using the Quotient Property of Radicals knowledge point

$$ \sqrt[4]{\frac{x}{x^2}} = \sqrt[4]{\frac{1}{x}} $$

Answer:

No, it is still possible to write the expression as a single radical. By converting the radicals to rational exponents, finding a common denominator for the fractional exponents, and rewriting them with a common index, the quotient property can then be applied to combine them into a single radical:

$$ \frac{\sqrt[4]{x}}{\sqrt{x}} = \frac{x^{1/4}}{x^{2/4}} = \sqrt[4]{\frac{x}{x^2}} = \sqrt[4]{\frac{1}{x}} $$