Simplify square root of cube root of square root of y

Solution:

Step 1

Express the cube root of the square root of y as y raised to the power of one-sixth. Thus, we have $\sqrt[3]{\sqrt{y}} = \sqrt[6]{y}$. Now, consider the square root of this expression, which gives us $\sqrt{\sqrt[6]{y}}$.

Step 2

Combine the square root and the sixth root into a single radical expression by multiplying the indices, resulting in the twelfth root of y. Therefore, $\sqrt{\sqrt[6]{y}} = \sqrt[12]{y}$.

Knowledge Notes:

The problem involves nested radicals, which means a radical (square root, cube root, etc.) is within another radical. To simplify nested radicals, we can use the property that a radical can be expressed as a power with a fractional exponent, where the numerator is the power and the denominator is the index of the root. For instance, $\sqrt[n]{x^m} = x^{\frac{m}{n}}$.

When simplifying nested radicals, we can combine the radicals by multiplying their fractional exponents. If we have $\sqrt[n]{\sqrt[m]{x}}$, this can be written as $x^{\frac{1}{n} \cdot \frac{1}{m}} = x^{\frac{1}{n \cdot m}}$. This is the principle used in the solution process.

In this specific problem, we have a square root of a cube root of a square root. The square root is the same as raising to the power of $\frac{1}{2}$, and the cube root is the same as raising to the power of $\frac{1}{3}$. When we combine these, we multiply the fractional exponents to get the final simplified form.