Can you teach me how to convert mixed radical into entire radicals and vice versa?

Sure! Let's break down how to convert between mixed radicals and entire radicals.

### 1. Converting a **Mixed Radical** to an **Entire Radical**

A mixed radical is a radical expression that has a coefficient (number outside the square root). For example, \( 3\sqrt{2} \) is a mixed radical.

To convert a mixed radical to an entire radical:
1. **Square the coefficient** and place it inside the radical.
2. Multiply it by the number already inside the radical.

**Example:**
Convert \( 3\sqrt{2} \) into an entire radical.
- Step 1: Square the coefficient \( 3 \), which gives \( 3^2 = 9 \).
- Step 2: Multiply \( 9 \) by the number inside the radical: \( 9 \times 2 = 18 \).

So, \( 3\sqrt{2} \) becomes \( \sqrt{18} \).

### 2. Converting an **Entire Radical** to a **Mixed Radical**

An entire radical is a radical expression without a coefficient outside the square root. For example, \( \sqrt{50} \) is an entire radical.

To convert an entire radical to a mixed radical:
1. Find the largest perfect square that divides the number inside the radical.
2. Take the square root of the perfect square and place it outside the radical as the coefficient.
3. Keep the remaining part inside the radical.

**Example:**
Convert \( \sqrt{50} \) into a mixed radical.
- Step 1: Find the largest perfect square that divides 50. The largest perfect square is \( 25 \), because \( 50 = 25 \times 2 \).
- Step 2: Take the square root of 25, which is \( 5 \).
- Step 3: Rewrite the radical: \( \sqrt{50} = 5\sqrt{2} \).

So, \( \sqrt{50} \) becomes \( 5\sqrt{2} \).

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### Summary of Steps:
- **Mixed Radical to Entire Radical**: Square the coefficient and multiply it by the number inside the radical.
- **Entire Radical to Mixed Radical**: Factor out the largest perfect square and simplify.

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Would you like more details or have any questions?

Here are 5 questions to expand your understanding:
1. How do you identify the largest perfect square when simplifying entire radicals?
2. Can you convert \( 2\sqrt{7} \) into an entire radical?
3. How would you simplify \( \sqrt{75} \) into a mixed radical?
4. What happens when the number inside the radical is already a perfect square?
5. Can you express \( 4\sqrt{5} \) as an entire radical?

**Tip:** Always check for the largest perfect square divisor to simplify entire radicals most efficiently.

Learn how to convert mixed radicals like 3√2 into entire radicals, and entire radicals like √50 into mixed radicals. This guide covers key algebraic concepts, such as simplifying radicals, properties of square roots, and recognizing perfect squares. Suitable for students in Grades 8-10.

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