https://file.aiquickdraw.com/mathbot/file/1726378563929pp0wx5sj.jpg

We are given that:

\[
\sqrt[3]{3668} = 15.422 \quad \text{and} \quad \sqrt[3]{36680} = 33.225
\]

The task is to find the value of \(\sqrt[3]{36.68}\).

Let's break it down.

### Step 1: Simplify \(36.68\)

We notice that \(36.68\) is a factor of \(3668\), specifically:

\[
36.68 = \frac{3668}{100}
\]

Thus, the cube root of \(36.68\) is related to the cube root of \(3668\).

### Step 2: Apply cube root property

We know that:

\[
\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}
\]

In this case, \(a = 3668\) and \(b = 100\), so:

\[
\sqrt[3]{36.68} = \frac{\sqrt[3]{3668}}{\sqrt[3]{100}}
\]

### Step 3: Calculate \(\sqrt[3]{100}\)

The cube root of \(100\) is:

\[
\sqrt[3]{100} \approx 4.6416
\]

### Step 4: Substitute values

Now, we substitute the values:

\[
\sqrt[3]{36.68} = \frac{15.422}{4.6416} \approx 3.323
\]

Thus, the value of \(\sqrt[3]{36.68}\) is approximately **3.323**.

---

Would you like further details or have any questions? Here are 5 related questions for deeper understanding:

1. How do you calculate the cube root of a fraction?
2. Why is the cube root of 100 approximately 4.6416?
3. Can you verify this approximation using logarithms?
4. What are the properties of cube roots that simplify expressions like \(\sqrt[3]{a/b}\)?
5. How would you calculate \(\sqrt[3]{36.68}\) using a calculator?

**Tip:** When simplifying cube roots of decimal numbers, it's often helpful to express them as fractions for easier computation.

Given that ∛3668 = 15.422 and ∛36680 = 33.225, find the value of ∛36.68.

Learn how to calculate the cube root of 36.68 using given values of cube roots for 3668 and 36680. This problem explores key algebraic concepts such as cube root properties and simplifying fractions. Suitable for students in Grades 8-10.

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