\sqrt[]{\frac{ \sqrt[3]{64}+\sqrt[4]{256}}{\sqrt[]{64}+\sqrt[]{256}}}

Let's simplify the given expression step by step:

\[
\sqrt{\frac{\sqrt[3]{64} + \sqrt[4]{256}}{\sqrt{64} + \sqrt{256}}}
\]

### Step 1: Simplify the roots individually.
1. \(\sqrt[3]{64}\):
   - \(64 = 4^3\), so \(\sqrt[3]{64} = 4\).

2. \(\sqrt[4]{256}\):
   - \(256 = 4^4\), so \(\sqrt[4]{256} = 4\).

3. \(\sqrt{64}\):
   - \(64 = 8^2\), so \(\sqrt{64} = 8\).

4. \(\sqrt{256}\):
   - \(256 = 16^2\), so \(\sqrt{256} = 16\).

### Step 2: Substitute the simplified values back into the expression.
The expression becomes:

\[
\sqrt{\frac{4 + 4}{8 + 16}} = \sqrt{\frac{8}{24}} = \sqrt{\frac{1}{3}}.
\]

### Step 3: Simplify the square root.
\[
\sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}}.
\]

### Final Answer:
\[
\boxed{\frac{1}{\sqrt{3}}}.
\]

Would you like any further details or have any questions?

Here are some related questions to expand on this:
1. What are cube roots and how are they different from square roots?
2. How can you rationalize a denominator containing a square root?
3. What is the general process for simplifying expressions with multiple roots?
4. How do fractional exponents relate to roots like \(\sqrt[3]{64}\)?
5. How can simplifying roots help in solving complex algebraic equations?

**Tip:** To simplify expressions faster, always start by breaking down each root into its prime factors!

\sqrt{\frac{ \sqrt[3]{64}+\sqrt[4]{256}}{\sqrt{64}+\sqrt{256}}}

Learn how to simplify the complex expression \sqrt{\frac{ \sqrt[3]{64}+\sqrt[4]{256}}{\sqrt{64}+\sqrt{256}}} with a detailed step-by-step solution. This problem covers key mathematical concepts related to roots and radicals, and is suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation