Simplify 260652 square root of 348

Solution:

Step 1:

Express $348$ as the product of its prime factors: $2^2 \cdot 87$.

Step 1.1:

Extract the square root of $4$ from $348$: $260652 \sqrt{4 \cdot 87}$.

Step 1.2:

Replace $4$ with $2^2$: $260652 \sqrt{2^2 \cdot 87}$.

Step 2:

Remove the perfect square from under the radical sign: $260652 \cdot (2 \sqrt{87})$.

Step 3:

Perform the multiplication of $2$ and $260652$: $521304 \sqrt{87}$.

Step 4:

Present the result in its various forms:

• Exact Form: $521304 \sqrt{87}$
• Decimal Form: $4862400.00989141$

Knowledge Notes:

The problem involves simplifying a radical expression. Here are the relevant knowledge points and detailed explanations:

1. Prime Factorization: This is the process of breaking down a composite number into its prime factors. For instance, $348$ can be factored into $2^2 \cdot 87$, where $2$ is a prime number.

2. Simplifying Square Roots: When simplifying square roots, any perfect square factors can be taken out of the radical. For example, $\sqrt{4 \cdot 87}$ can be simplified to $2\sqrt{87}$ because $4$ is a perfect square and its square root is $2$.

3. Multiplication Outside the Radical: After simplifying the radical, any numbers outside can be multiplied together. In this case, $260652$ is multiplied by $2$ to get $521304$.

4. Exact vs. Decimal Form: The exact form of a radical expression keeps the square root intact, while the decimal form represents the approximate value of the radical expression as a decimal number.

5. Use of LaTeX: In the solution, LaTeX is used to properly format mathematical expressions, ensuring that numbers, symbols, and operations are clearly presented.

By understanding these concepts, one can effectively simplify radical expressions and perform operations involving square roots.