Simplify square root of cube root of 36
The question is asking for the mathematical operation that simplifies the expression which involves taking the square root of the cube root of the number 36. You're being asked to perform a two-step nested radical simplification: the cube root (which is the 3rd root of a number) should be taken of 36 first, and then the square root (which is the 2nd root) of the resulting value from the first operation. Essentially, you'll be finding the 6th root of 36 by combining these two steps, but the question specifically wants the operations to be performed successively and the expression to be simplified as much as possible.
$\sqrt{\sqrt[3]{36}}$
Answer
Solution:
Simplification Process:
Step 1: Convert the nested radicals into a single radical with an exponent that represents the product of the indices. $\sqrt{\sqrt[3]{36}}$ becomes $\sqrt[6]{36}$.
Step 2: Express the number under the radical in its prime factorized form. In this case, $36$ is $6^2$. So, we have $\sqrt[6]{6^2}$.
Step 3: Reorganize the radical to separate the two different roots. This gives us $\sqrt[3]{\sqrt{6^2}}$.
Step 4: Simplify the expression by extracting roots where possible. Since the square root of $6^2$ is $6$, we get $\sqrt[3]{6}$.
Step 5: Present the simplified result in both exact and decimal forms. The exact form is $\sqrt[3]{6}$, and the decimal approximation is $1.81712059 \ldots$.
Knowledge Notes:
To simplify nested radicals, one must understand the properties of exponents and radicals. Here are the relevant knowledge points:
Nested Radicals: A nested radical is a radical expression within another radical expression. In this problem, we have a square root nested within a cube root.
Radical Exponents: The nth root of a number can be expressed as that number raised to the power of 1/n. Therefore, $\sqrt{x} = x^{1/2}$ and $\sqrt[3]{x} = x^{1/3}$.
Combining Radicals: When you have a radical within another radical, you can multiply the indices to combine them into a single radical. For example, $\sqrt{\sqrt[3]{x}} = \sqrt[6]{x}$.
Prime Factorization: Expressing a number as the product of its prime factors can simplify the process of taking roots. For instance, $36 = 6^2$.
Simplifying Radicals: When the index of the radical and the exponent of the number inside have a common factor, simplification is possible by extracting the root. For example, $\sqrt[6]{6^2}$ simplifies to $\sqrt[3]{6}$.
Exact and Decimal Forms: The exact form of a radical expression is left in terms of roots (e.g., $\sqrt[3]{6}$), while the decimal form is a numerical approximation (e.g., $1.81712059 \ldots$).
Understanding these concepts is crucial for simplifying radical expressions and solving problems involving roots.
