Go step by step to reduce the radical., This question page includes a given radical, a line to fill out of the form square root of, input box, end square root, square root of, input box, end square root, and a try button. After pressing the try button, if your answer is correct, it will be added to the page and a new step will appear with another try button. Continue this process until the radical is fully reduced and the submit answer button appears.

square root of, 192, end square root192​

square root of, end square root *x* ​square root of, end square root *x*​ try


Let's go through the process of reducing the radical step by step, using the radical \(\sqrt{192}\) as an example.

### Step 1: Factor the number inside the square root
First, we need to factor 192 into its prime factors.

\[
192 = 2 \times 96 \\
96 = 2 \times 48 \\
48 = 2 \times 24 \\
24 = 2 \times 12 \\
12 = 2 \times 6 \\
6 = 2 \times 3
\]

So, \(192\) can be factored as:

\[
192 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3
\]

This can be rewritten as:

\[
192 = 2^6 \times 3
\]

### Step 2: Simplify the square root
Now, we simplify the square root by taking the square root of the perfect square part (i.e., \(2^6\)).

\[
\sqrt{192} = \sqrt{2^6 \times 3}
\]

We know that \(\sqrt{2^6}\) simplifies to \(2^3\) because \(\sqrt{(2^6)} = 2^{6/2} = 2^3\).

So, the expression simplifies to:

\[
\sqrt{192} = 2^3 \times \sqrt{3} = 8\sqrt{3}
\]

### Step 3: Input and Try
In the system you're describing:

1. **First input box**: After recognizing that \(\sqrt{192} = 8\sqrt{3}\), you will first type `8` in the first input box, indicating that the coefficient of the simplified radical is 8.
   
2. **Second input box**: In the next step, you input `3` in the second input box, representing the remaining square root, \(\sqrt{3}\).

After entering these values, pressing the "Try" button should confirm that \( \sqrt{192} \) simplifies to \( 8\sqrt{3} \).

If correct, a new step might appear, but in this case, the radical is fully reduced, so you would press the "Submit Answer" button.

Would you like to go over any part of this process in more detail or practice with another example?

Here are five related questions:
1. How do you identify perfect squares in a number?
2. What are the steps to factor numbers into their prime factors?
3. How do you simplify radicals with variables?
4. What is the significance of prime factorization in simplifying radicals?
5. Can you apply these steps to a radical that is a cube root instead?

**Tip:** Always factor the number under the radical completely to ensure you can extract the largest perfect square.

Learn how to reduce the radical step by step, using the example of simplifying \(\sqrt{192}\). This process involves square roots, prime factorization, and radical simplification. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation